$SU(1,1, \mathbb{C}) \simeq SL(2, \mathbb{R})$ as they preserve similar quadratic forms
- $|x|^2 -| y|^2=1$
- $ad-bc=1$
- let $x+y= a+bi$ and $x-y= c+di$
if instead of $ \mathbb{R}$ try a ring of integers $\mathcal{O}_K$ these are Bianchi groups. I am not am expert. the case of $\mathcal{O}_K=\mathbb{Z}[i]$ has explicit generators as found by Hurwitz
- the general case looks wide open. here is paper of swan
- Bianchi original notes (in Italian) can be found on Wikipedia and make for interesting reading. [1]
- the Ford circles and Farey fractions are relevant objects.
- if you google "Ford fractions" Ford's original discussion comes up. at the very end he briefly discusses $\mathbb{Z}[i]$ using spheres of various fractional radii. And the action of the Bianchi group.
I find it utterly mind-boggling there is no easy extension of Farey fractions to general quadratic rings $\mathcal{O}_K$ with $K=\mathbb{Q}(\sqrt{-D})$. It's non-trivial