Isomorphism between Gaussian Integer Units and Integers Modulo 4 under Addition

Theorem
Let $\struct {U_\C, \times}$ be the group of Gaussian integer units under complex multiplication.

Let $\struct {\Z_n, +_4}$ be the integers modulo $4$ under modulo addition.

Then $\struct {U_\C, \times}$ and $\struct {\Z_4, +_4}$ are isomorphic algebraic structures.