The group $SL_2(\mathbb{Z})$ contains many free subgroups, for example all of the principal congruence subgroups for $n\geq 3$ and the subgroup $\left\langle \left(\begin{array}{cc}
1 & 2\\
0 & 1
\end{array}\right),\left(\begin{array}{cc}
1 & 0\\
2 & 1
\end{array}\right)\right\rangle $ which is almost the principal congruence subgroup for n=2, and more over it has a finite index in $SL_2(\mathbb{Z})$. In addition, when projecting $SL_2(\mathbb{Z})$ onto $SL_2(\mathbb{Z}/p)$, the restriction to these subgroups is also surjective as long as $p \nmid n$.

Is a similar phenomena true for $SL_2(\mathbb{Z}[i])$, or more generally when we substitute $\mathbb{Z}$ with the ring of integers $\mathcal{O}_k$ for some number field $k$?