@chwala ,
You may find it more direct to find a unit (other than ##1,\,-1##) by finding the multiplicative inverse of the general element, ##\displaystyle a+b\sqrt 3 ##, and tailoring the values of ##a \text{ and } b## as needed.
Again, working in the ring ##\displaystyle \mathbb{Z}[\sqrt{3}] \,## .
##\displaystyle \dfrac{1}{a+b\sqrt 3 } = \dfrac{(a-b\sqrt 3 \,)}{(a+b\sqrt 3 \,)(a-b\sqrt 3 \,)} = \dfrac{(a-b\sqrt 3 \,)}{a^2-3b^2}##
It seems to me that the denominator needs to be ##\pm 1##. It certainly would be convenient if we could find values for ##a## and ##b## which make it ##1## . Finding such values would mean that the inverse of the element is the element's conjugate .
We want ##\displaystyle a^2-3b^2 = 1## .
Borrowing the idea from
@fresh_42 , let's look at this modulo ##3##.
##\displaystyle a^2\equiv 1\pmod{3} ##.
##a^2 = 1## doesn't appear to work well. It just gives ##b=0## .
Trying ##a^2 = 4## gives ##\displaystyle 4-3b^2 = 1## , so ##b=\pm1##.
So we have that ##\displaystyle (2+1\sqrt 3 \,)(2-1\sqrt 3 \,) = 1##
Both ##(2+1\sqrt 3) \text { and } (2-1\sqrt 3)## are units in ##\displaystyle \mathbb{Z}[\sqrt{3}\,] \,##.