$SU(1,1, \mathbb{C}) \simeq SL(2, \mathbb{R})$ 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](http://www.sciencedirect.com/science/article/pii/0001870871900272)
* Bianchi original notes (in Italian) can be found on [Wikipedia](https://en.m.wikipedia.org/wiki/Bianchi_group) and make for interesting reading.