Gaussian integers: Proof that if $N(\alpha) = N(\beta)$ then $\alpha = t \beta, t \in \{1, -1, i, -i \}$

Question

I need to prove that for $\alpha, \beta$ Gaussian integers: $$N(\alpha) = N(\beta)\Leftrightarrow \alpha = t\beta, t \in \{-1, 1, i, -i\}$$ Now I have found out that if $\alpha = t\beta \Rightarrow N(\alpha) = N(t \beta) = N(t)N(\beta) = 1 N(\beta) = N(\beta).$

How do I prove the other way around?