Can a Gaussian integer matrix have an inverse with Gaussian integer entries? Is there any way to characterize the set of complex matrices with Gaussian integer entries whose inverses also have Gaussian integer entries? I'm aware of the numerous examples of integer matrices whose inverses also have integer entries (usually involving binomial coefficients), but I'm wondering if those constructions can be generalized to Gaussian integers.

If $R$ is a subring of $\Bbb C$, the group ${\rm GL}_n(R)$ of the invertible matrices with coefficients in $R$ consists of the matrices with determinant in $R^\times$ (the group of invertible elements in $R$).
Thus, if $R={\Bbb Z}[i]$ is the ring of Gaussian integers, the group ${\rm GL}_n(R)$ consists of the matrices $M$ with coefficients in $R$ such that
$$
\det(M)\in\{\pm1,\pm i\}.
$$
