While trying to prove that if $a^2+b^2$ is prime in $\mathbb{Z}[i]$ then $a+bi$ is also prime, I start solution by defining a norm $N$: $\mathbb{Z}[i] \to \mathbb{Z}_+$ as $N(a+bi)=a^2+b^2$.
But my teacher consider it wrong by saying that we can't define this map on sets other then Euclidean Domain.
And I don't understand why?
You're in the right direction despite what your teacher says.
You now need to prove these properties for $N$:
$N(\alpha \beta) = N(\alpha) N(\beta)$
$N(\alpha) = 1$ iff $\alpha$ is a unit in $\mathbb Z[i]$
Finally, make sure you see how what you want to prove follows from these properties.