Isomorphism between Gaussian Integer Units and Integers Modulo 4 under Addition

Theorem
Let $U_\C$ be the set of units of the Gaussian integers:
 * $U_\C = \left\{{1, i, -1, -i}\right\}$

where $i = \sqrt {-1}$ is the imaginary unit.

Let $\left({U_\C, \times}\right)$ be the set $U_\C$ under complex multiplication.

Let $\left({\Z_n, +_4}\right)$ be the integers modulo $4$ under modulo addition.

Then $\left({U_\C, \times}\right)$ and $\left({\Z_4, +_4}\right)$ are isomorphic algebraic structures.