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$SU(1,1, \mathbb{C}) \simeq SL(2, \mathbb{R})$ 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.