I think this answers your question
In[1]:= Reduce[c (q*z+(-1+q) (-1+Sqrt[1+q*z]))==d*(h*q^2 (1+z)^2),q]
From In[1]:= Reduce::useq: The answer found by Reduce contains unsolved equation(s) A likely reason for this is that the solution set depends on branch cuts of Mathematica functions.
Out[1]= (z == -1 && c == 0) || (z == 0 && h == 0 && c == 0) || (c != 0 &&
z == -1 && q == 0) || c != 0 && z == -1 && q == (-1 - Sqrt[5])/2) || (c*(-1
+ q) != 0 && z == 0 && h == 0) || (d*h*(1 + z) != 0 && -c != 0 && q == 0) || (z
+ z^2 != 0 && h == 0 && c == 0) || (h*z + h*z^2 != 0 && d == 0 && c == 0) ||
(d*h*z + d*h*z^2 != 0 && c == 0 && q == 0) || (c != 0 && z == 0 && h == 0 && q
== 1) || (h != 0 && z == 0 && d == 0 && c == 0) || (d*h != 0 && z == 0 && c == 0
&& q == 0) || (c*(-1 + q) != 0 && h != 0 && z == 0 && d == 0) || (z != 0 && d*h
+ 2*d*h*z + d*h*z^2 != 0 && c == (d*h + 2*d*h*z + d*h*z^2)/z && q == 1) || (c*z
!= 0 && 1 + z != 0 && h == 0 && q == 0) || (c*z != 0 && h + h*z != 0 && d == 0
&& q == 0) || (d*h*(1 + z) != 0 && c*(-1 + Root[c^2*z - 2*c*d*h*#1 -
4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h* #1^2 - c^2*z*#1^2 +
2*c*d*h*z*#1^2 - 2*c*d*h*z^2*#1^2 - 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 +
4*d^2*h^2*z* #1^3 + 6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 &
, 1]) != 0 && 0 == (-c + c*Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2*#1 -
2*c*d*h*z^2*#1 + 2*c*d*h* #1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h*z^2* #1^2
- 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z* #1^3 + 6*d^2*h^2*z^2*#1^3 +
4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 & , 1] - c*z*Root[c^2*z - 2*c*d*h*#1 -
4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h* #1^2 - c^2*z*#1^2 +
2*c*d*h*z*#1^2 - 2*c*d*h*z^2* #1^2 - 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 +
4*d^2*h^2*z* #1^3 + 6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 &
, 1] + d*h*Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1
+ 2*c*d*h* #1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h* z^2*#1^2 - 2*c*d*h*z^3*
#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z*#1^3 + 6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3
+ d^2*h^2*z^4*#1^3 & , 1]^2 + 2*d*h*z* Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 +
c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h* #1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 -
2*c*d*h* z^2*#1^2 - 2*c*d*h*z^3* #1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z*#1^3 +
6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 & , 1]^2 + d*h*z^2*
Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h*
#1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h* z^2*#1^2 - 2*c*d*h*z^3* #1^2 +
d^2*h^2*#1^3 + 4*d^2*h^2*z*#1^3 + 6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3 +
d^2*h^2*z^4*#1^3 & , 1]^2 + c*Sqrt[1 + z*Root[ c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1
+ c^2*z^2* #1 - 2*c*d*h*z^2*#1 + 2*c*d*h*#1^2 - c^2*z* #1^2 + 2*c*d*h*z*#1^2 -
2*c*d*h*z^2*#1^2 - 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2* z*#1^3 +
6*d^2*h^2*z^2* #1^3 + 4*d^2*h^2*z^3* #1^3 + d^2*h^2*z^4* #1^3 & , 1]] -
c*Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 +
2*c*d*h* #1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h*z^2* #1^2 -
2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z* #1^3 + 6*d^2*h^2*z^2*#1^3 +
4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 & , 1]* Sqrt[1 + z*Root[ c^2*z -
2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2* #1 - 2*c*d*h*z^2*#1 + 2*c*d*h*#1^2 - c^2*z*
#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h*z^2*#1^2 - 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 +
4*d^2*h^2* z*#1^3 + 6*d^2*h^2*z^2* #1^3 + 4*d^2*h^2*z^3* #1^3 + d^2*h^2*z^4*
#1^3 & , 1]])/ (c*(-1 + Root[c^2*z - 2*c*d*h* #1 - 4*c*d*h*z*#1 + c^2*z^2*#1 -
2*c*d*h*z^2* #1 + 2*c*d*h*#1^2 - c^2*z*#1^2 + 2*c*d*h*z* #1^2 - 2*c*d*h*z^2*#1^2
- 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z* #1^3 + 6*d^2*h^2*z^2*#1^3 +
4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 & , 1])) && q == Root[c^2*z - 2*c*d*h*#1 -
4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h*#1^2 - c^2*z*#1^2 +
2*c*d*h*z*#1^2 - 2*c*d*h*z^2*#1^2 - 2*c*d*h*z^3* #1^2 + d^2*h^2*#1^3 +
4*d^2*h^2*z*#1^3 + 6*d^2*h^2* z^2*#1^3 + 4*d^2*h^2*z^3* #1^3 + d^2*h^2*z^4*#1^3
& , 1]) || (d*h*(1 + z) != 0 && c*(-1 + Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 +
c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h* #1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 -
2*c*d*h*z^2* #1^2 - 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z* #1^3 +
6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 & , 2]) != 0 && 0 ==
(-c + c*Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 +
2*c*d*h* #1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h*z^2* #1^2 -
2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z* #1^3 + 6*d^2*h^2*z^2*#1^3 +
4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 & , 2] - c*z*Root[c^2*z - 2*c*d*h*#1 -
4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h* #1^2 - c^2*z*#1^2 +
2*c*d*h*z*#1^2 - 2*c*d*h*z^2* #1^2 - 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 +
4*d^2*h^2*z* #1^3 + 6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 &
, 2] + d*h*Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1
+ 2*c*d*h* #1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h* z^2*#1^2 - 2*c*d*h*z^3*
#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z*#1^3 + 6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3
+ d^2*h^2*z^4*#1^3 & , 2]^2 + 2*d*h*z* Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 +
c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h* #1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 -
2*c*d*h* z^2*#1^2 - 2*c*d*h*z^3* #1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z*#1^3 +
6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 & , 2]^2 + d*h*z^2*
Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h*
#1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h* z^2*#1^2 - 2*c*d*h*z^3* #1^2 +
d^2*h^2*#1^3 + 4*d^2*h^2*z*#1^3 + 6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3 +
d^2*h^2*z^4*#1^3 & , 2]^2 + c*Sqrt[1 + z*Root[ c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1
+ c^2*z^2* #1 - 2*c*d*h*z^2*#1 + 2*c*d*h*#1^2 - c^2*z* #1^2 + 2*c*d*h*z*#1^2 -
2*c*d*h*z^2*#1^2 - 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2* z*#1^3 +
6*d^2*h^2*z^2* #1^3 + 4*d^2*h^2*z^3* #1^3 + d^2*h^2*z^4* #1^3 & , 2]] -
c*Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 +
2*c*d*h* #1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h*z^2* #1^2 -
2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z* #1^3 + 6*d^2*h^2*z^2*#1^3 +
4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 & , 2]* Sqrt[1 + z*Root[ c^2*z -
2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2* #1 - 2*c*d*h*z^2*#1 + 2*c*d*h*#1^2 - c^2*z*
#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h*z^2*#1^2 - 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 +
4*d^2*h^2* z*#1^3 + 6*d^2*h^2*z^2* #1^3 + 4*d^2*h^2*z^3* #1^3 + d^2*h^2*z^4*
#1^3 & , 2]])/ (c*(-1 + Root[c^2*z - 2*c*d*h* #1 - 4*c*d*h*z*#1 + c^2*z^2*#1 -
2*c*d*h*z^2* #1 + 2*c*d*h*#1^2 - c^2*z*#1^2 + 2*c*d*h*z* #1^2 - 2*c*d*h*z^2*#1^2
- 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z* #1^3 + 6*d^2*h^2*z^2*#1^3 +
4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 & , 2])) && q == Root[c^2*z - 2*c*d*h*#1 -
4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h*#1^2 - c^2*z*#1^2 +
2*c*d*h*z*#1^2 - 2*c*d*h*z^2*#1^2 - 2*c*d*h*z^3* #1^2 + d^2*h^2*#1^3 +
4*d^2*h^2*z*#1^3 + 6*d^2*h^2* z^2*#1^3 + 4*d^2*h^2*z^3* #1^3 + d^2*h^2*z^4*#1^3
& , 2]) || (d*h*(1 + z) != 0 && c*(-1 + Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 +
c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h* #1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 -
2*c*d*h*z^2* #1^2 - 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z* #1^3 +
6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 & , 3]) != 0 && 0 ==
(-c + c*Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 +
2*c*d*h* #1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h*z^2* #1^2 -
2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z* #1^3 + 6*d^2*h^2*z^2*#1^3 +
4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 & , 3] - c*z*Root[c^2*z - 2*c*d*h*#1 -
4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h* #1^2 - c^2*z*#1^2 +
2*c*d*h*z*#1^2 - 2*c*d*h*z^2* #1^2 - 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 +
4*d^2*h^2*z* #1^3 + 6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 &
, 3] + d*h*Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1
+ 2*c*d*h* #1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h* z^2*#1^2 - 2*c*d*h*z^3*
#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z*#1^3 + 6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3
+ d^2*h^2*z^4*#1^3 & , 3]^2 + 2*d*h*z* Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 +
c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h* #1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 -
2*c*d*h* z^2*#1^2 - 2*c*d*h*z^3* #1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z*#1^3 +
6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 & , 3]^2 + d*h*z^2*
Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h*
#1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h* z^2*#1^2 - 2*c*d*h*z^3* #1^2 +
d^2*h^2*#1^3 + 4*d^2*h^2*z*#1^3 + 6*d^2*h^2*z^2*#1^3 + 4*d^2*h^2*z^3*#1^3 +
d^2*h^2*z^4*#1^3 & , 3]^2 + c*Sqrt[1 + z*Root[ c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1
+ c^2*z^2* #1 - 2*c*d*h*z^2*#1 + 2*c*d*h*#1^2 - c^2*z* #1^2 + 2*c*d*h*z*#1^2 -
2*c*d*h*z^2*#1^2 - 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2* z*#1^3 +
6*d^2*h^2*z^2* #1^3 + 4*d^2*h^2*z^3* #1^3 + d^2*h^2*z^4* #1^3 & , 3]] -
c*Root[c^2*z - 2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 +
2*c*d*h* #1^2 - c^2*z*#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h*z^2* #1^2 -
2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z* #1^3 + 6*d^2*h^2*z^2*#1^3 +
4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 & , 3]* Sqrt[1 + z*Root[ c^2*z -
2*c*d*h*#1 - 4*c*d*h*z*#1 + c^2*z^2* #1 - 2*c*d*h*z^2*#1 + 2*c*d*h*#1^2 - c^2*z*
#1^2 + 2*c*d*h*z*#1^2 - 2*c*d*h*z^2*#1^2 - 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 +
4*d^2*h^2* z*#1^3 + 6*d^2*h^2*z^2* #1^3 + 4*d^2*h^2*z^3* #1^3 + d^2*h^2*z^4*
#1^3 & , 3]])/ (c*(-1 + Root[c^2*z - 2*c*d*h* #1 - 4*c*d*h*z*#1 + c^2*z^2*#1 -
2*c*d*h*z^2* #1 + 2*c*d*h*#1^2 - c^2*z*#1^2 + 2*c*d*h*z* #1^2 - 2*c*d*h*z^2*#1^2
- 2*c*d*h*z^3*#1^2 + d^2*h^2*#1^3 + 4*d^2*h^2*z* #1^3 + 6*d^2*h^2*z^2*#1^3 +
4*d^2*h^2*z^3*#1^3 + d^2*h^2*z^4*#1^3 & , 3])) && q == Root[c^2*z - 2*c*d*h*#1 -
4*c*d*h*z*#1 + c^2*z^2*#1 - 2*c*d*h*z^2*#1 + 2*c*d*h*#1^2 - c^2*z*#1^2 +
2*c*d*h*z*#1^2 - 2*c*d*h*z^2*#1^2 - 2*c*d*h*z^3* #1^2 + d^2*h^2*#1^3 +
4*d^2*h^2*z*#1^3 + 6*d^2*h^2* z^2*#1^3 + 4*d^2*h^2*z^3* #1^3 + d^2*h^2*z^4*#1^3
& , 3]) || (c != 0 && h != 0 && z == 0 && d == 0 && q == 1) || (c*z != 0 && -2 +
z - Sqrt[4 + z^2] != 0 && 0 == (4 - 2*z + 2*z^2 + 2*Sqrt[4 + z^2] - 2*z*Sqrt[4 +
z^2] - 2*Sqrt[2]*Sqrt[2 + z^2 - z*Sqrt[4 + z^2]] + Sqrt[2]*z*Sqrt[2 + z^2 -
z*Sqrt[4 + z^2]] - Sqrt[2]*Sqrt[4 + z^2]* Sqrt[2 + z^2 - z*Sqrt[4 + z^2]])/
(2*(2 - z + Sqrt[4 + z^2])) && 1 + z != 0 && h == 0 && q == (z - Sqrt[4 +
z^2])/2) || (c*z != 0 && -2 + z - Sqrt[4 + z^2] != 0 && 0 == (4 - 2*z + 2*z^2 +
2*Sqrt[4 + z^2] - 2*z*Sqrt[4 + z^2] - 2*Sqrt[2]*Sqrt[2 + z^2 - z*Sqrt[4 + z^2]]
+ Sqrt[2]*z*Sqrt[2 + z^2 - z*Sqrt[4 + z^2]] - Sqrt[2]*Sqrt[4 + z^2]* Sqrt[2 +
z^2 - z*Sqrt[4 + z^2]])/ (2*(2 - z + Sqrt[4 + z^2])) && h + h*z != 0 && d == 0
&& q == (z - Sqrt[4 + z^2])/2) || (c*z != 0 && -2 + z + Sqrt[4 + z^2] != 0 && 0
== (-4 + 2*z - 2*z^2 + 2*Sqrt[4 + z^2] - 2*z*Sqrt[4 + z^2] + 2*Sqrt[2]*Sqrt[2 +
z^2 + z*Sqrt[4 + z^2]] - Sqrt[2]*z*Sqrt[2 + z^2 + z*Sqrt[4 + z^2]] -
Sqrt[2]*Sqrt[4 + z^2]* Sqrt[2 + z^2 + z*Sqrt[4 + z^2]])/ (2*(-2 + z + Sqrt[4 +
z^2])) && 1 + z != 0 && h == 0 && q == (z + Sqrt[4 + z^2])/2) || (c*z != 0 && -2
+ z + Sqrt[4 + z^2] != 0 && 0 == (-4 + 2*z - 2*z^2 + 2*Sqrt[4 + z^2] -
2*z*Sqrt[4 + z^2] + 2*Sqrt[2]*Sqrt[2 + z^2 + z*Sqrt[4 + z^2]] - Sqrt[2]*z*Sqrt[2
+ z^2 + z*Sqrt[4 + z^2]] - Sqrt[2]*Sqrt[4 + z^2]* Sqrt[2 + z^2 + z*Sqrt[4 +
z^2]])/ (2*(-2 + z + Sqrt[4 + z^2])) && h + h*z != 0 && d == 0 && q == (z +
Sqrt[4 + z^2])/2)
So, for example, if you look at the first line of the "solution" you see
(z == -1 && c == 0) || lots_of_other_stuff.
Thus if we substitute -1 for z and 0 for c into your original equation we get
c (q*z + (-1 + q) (-1 + Sqrt[1 + q*z])) == d*(h*q^2 (1 + z)^2)
0 (q*z + (-1 + q) (-1 + Sqrt[1 + q* -1])) == d*(h*q^2 (1 + -1)^2)
0 (q*z + (-1 + q) (-1 + Sqrt[1 + q* -1])) == d*(h*q^2 0^2)
0 == 0
And likewise for all the other alternatives, some of which are very complicated.
Buried inside that are some
Root[expression_containing_#,n]
That is a much simpler way of saying that this is the nth root of the expression.
For your problem it appears possible to expand all those Root[] out to the full result containing radicals, but expands the size of this by several fold.
If you have any additional information, like whether some of your variables are or are not equal to zero, whether they are positive, whether they are or are not equal to each other, or that 1+q*z will be greater than or equal to zero so that the square root will not be complex, etc., then it might be possible to substantially simplify that. But there are no guarantees.
Unless you can provide additional information or you can identify one of the alternatives that captures your problem, both those probably being close to the same thing, it looks like such a "solution" probably isn't going to be of any use to you.
Simplify[] can sometimes reduce the size of things like this by perhaps half, but sometimes that will lose a solution.
