This may be a naïve question, but I have been thinking about the structure of the the matrix group $SL_2(\mathbb{Z}[i]) \subset SL_2(\mathbb{C})$. One thing that has been on my mind is whether or not $SL_2(\mathbb{Z}[i])$ is finitely generated. We know that $SL_2(\mathbb{Z}) \subset SL_2(\mathbb{R})$ is finitely generated, namely by $$S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}; \quad T =\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$
Since, $S$ and $T$ are in $SL_2(\mathbb{Z}[i])$ is it possible to come up with a generating set? My thought would be potentially having $S, T$ and maybe $i I_2$. If anyone has any insight, please share.
From a relatively elementary viewpoint (noting @MoisheKohan's observation in comments), it is indeed possible to prove the finite generation, by explicit matrices, in effect by explicit proof of unicuspidality of the quotient of hyperbolic three-space by this group. Namely, translations $\pmatrix{1&1\cr 0&1}$ and $\pmatrix{1& i\cr 0&1}$ generate all translations, and together with inversion $\pmatrix{0 & -1\cr 1& 0}$ (and maybe $\pmatrix{\pm i& 0\cr 0&\mp i}$ and $\pmatrix{-1&0\cr 0&-1}$, depending how we want to pose things), do succeed in moving every element in hyperbolic three-space into an explicit Siegel set... with good further details. This is done in some detail in chapter one of my CUP book(s) "Modern Analysis of Automorphic Forms, I, II"... also available (legally) on-line at www.math.umn.edu/~garrett/m/v/current_version.pdf