So I found an exercise to show that every non constant polynomial of $\mathbb{C}[X]$ is an open map. It goes as follows:
Let $P \in \mathbb{C}[x]$ be non constant.
First, we assume that $P(0) = 0$. Show that for all $r > 0$, there exists a $f(r) > 0$ such that $D(0, f(r)) \subset P(D(0, r))$ where we set, for $z \in \mathbb{C}$ and $r > 0$, $D(z, r) = \{ x \in \mathbb{C}, |x - z| < r \}$.
Then show that $P: \mathbb{C} \to \mathbb{C}$ is an open map (this time without assuming that $P(0) = 0$).
I managed to prove the "$D(0, f(r)) \subset P(D(0, r))$" thing. However, I do not see how this helps to prove what we want ...
Thank you for your help.