I am investigating solutions to Fermat's equation $x^n + y^n = z^n$ with $x,y,z$ in the Gaussian integers, excluding integers and pure imaginaries.
I have found out that there are only trivial solutions for the $n=3$ and $n=4$ cases, e.g. here.
I would be grateful if you let me know of the current status or if it is already a theorem.