It seems well understood that $a^4+b^4+c^4=d^4$ has non-trivial solutions in the integers. I found a parametrization of a class of solutions here $$a=85v^2+484v-313 \quad\quad b=68v^2-586v+10$$ $$ c=2\sqrt{22030+28849v-56158v^2+36941v^3-31790v^4} \quad d=357v^2-204v+363$$ which still seems incomplete since one must search through values of $v$ to find one that yields a rational $c$ like $v = \frac{-31}{467}$ for example.
I'd like to find a solution in the $\textbf{complex numbers}$ - so in the Gaussian integers specifically - with the added condition that $a$, $b$, and $c$ have the $\textbf{same length}$. So $$\|a\|=\|b\|=\|c\|\quad\text{for}\quad a,b,c\in \mathbb{Z}[i]$$
I tried some substitution like $$r=a+b+c \quad\quad s=ab+ac+bc \quad\quad t=abc$$ This yields $$a^4+b^4+c^4=(r^2-2s)^2-2(s^2-2rt)$$ But unfortunately this new equation - though very interesting - isn't any easier to solve. Further, $r,s,$ and $t$ don't even preserve any information about the constraint $\|a\|=\|b\|=\|c\|$
Does there exist a non-trivial solution to $\|a\|=\|b\|=\|c\|$ and $a^4+b^4+c^4=d^4$ for $a,b,c\in\mathbb{Z}[i]$?