Maybe I should have posted this as a comment to Gian Maria Dall'Ara very nice proof, because is a mere variation.

He uses the lemma : Any open proper map to a locally compact space surjects the connected components it reaches. Now any non-constant polynomial corestricted to its regular value locus is as in the lemma, so it is surjective.

Here is a "constructive proof" of the D'Alembert-Gauss theorem. Fix a degree $n > 0.$

Consider the $M_n$ the affine space of monic polynomials of degree $n$ and the proper map $\mathbb C^n \to M_n$, mapping a tuple $(z_1, \ldots , z_n)$ to $\Pi (X-z_i).$ Its critical values locus is non-disconnecting because it is a complex (singular) hypersurface : it is a polynomial image of the arrangement of hyperplanes { $z_i = z_j$}.

The map is therefore surjective.