Gaussian integer solutions for the fifth power taxicab number problem

The famous open problem of finding positive integer solutions to $a^5 + b^5 = c^5 + d^5$ is considered, and related solutions are found in two distinct settings: firstly, where $a$ and $b$ are both positive integers with $c$ and $d$ both Gaussian integers; secondly, where all of $a$, $b$, $c$, and $d$ are Gaussian integers.