Gaussian Integer Units are 4th Roots of Unity

Theorem
The units of the ring of Gaussian integers:
 * $\set {1, i, -1, -i}$

are the (complex) $4$th roots of $1$.

Proof
We have that $i = \sqrt {-1}$ is the imaginary unit.

Thus:

So $\set {1, i, -1, -i}$ constitutes the set of the $4$th roots of unity.