Isomorphism between Gaussian Integer Units and Integers Modulo 4 under Addition

Theorem
Let $\left({U_\C, \times}\right)$ be the group of Gaussian integer units 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.