Given a non-zero Gaussian integer $z$, we define the set $\mathcal S(z)$ containing all solutions of $ab+cd=z$ satisfying $\min(\vert a\vert,\vert b\vert)>\max(\vert c\vert,\vert d\vert)$ with $a,b,c,d$ in the set $\mathbb Z[i]$ of Gaussian integers.

The identity $$2m+1=(2n+(2n^2-m-1)i)(2n-(2n^2-m-1)i)-(2n^2-m)^2$$ and a similar identity for even integers implies that $\mathcal S(z)$ is infinite for every ordinary non-zero integer.

More generally $\mathcal S(z)$ is therefore infinite if $z$ is of the form $i^k s^2m$ for $s$ a non-zero Gaussian integer and for $m$ a non-zero ordinary integer.

Is there a computable bound on the size of solutions if $\mathcal S(z)$ is finite?

(The set $\mathcal S(1+i)$ for example seems to be finite).