Gaussian Erdős–Moser equation conjecture

For positive integers $k$ and $m$, let $\sigma_k(m-1)$ denote the sum of the $k$-th powers of the Gaussian integers in the relevant Gaussian-integer range, and consider the equation

$$\sigma_k(m-1)=(m+mi)^k.$$

Gaussian Erdős–Moser conjecture. The equation above has only the solution $(k,m)=(2,3)$:

$$(1+i)^2+(1+2i)^2+(2+i)^2+(2+2i)^2=18i=(3+3i)^2.$$

This is presented after computations for $k,m<100$ as a conjectural Gaussian analogue of the Erdős–Moser equation; no proof or resolution beyond those computations is given.