What is the Number of Complex Roots for the Given Equation with Varying d?

[eradi]

How many complex roots admit the following equation:
(2 z^2 + 1)^2 ((z + d)/(z - i))^1/2 - (2 z^2 - d)^2 ((z + i)/(z - d))^1/2 == 0
for 0 < d < 1, where i = (-1)^1/2.
Can I found how their number varies with d by using the argument principle?
Thanks in advance for helpfull suggestions

 

[Eynstone]

You get a fourth deg. equation in z^2 equation on simplification, so a maximum of 8 zeroes is possible.

 

[eradi]

Yes, but which of them are actually zeroes of the starting equation?
Thanks for your interest

 

[eradi]

I found numerically that for small d only four roots are admissible (two reals and two purely imaginary) but for large d two more complex conjugate roots appear.
There exists any analitical tools to define this critical value of d?