Here's an extract of my post to sci.math.research from 2001. The proof definitely uses a "sledgehammer method", but perhaps it has some pedagogical value. I have no doubt that other people may have come up with similar arguments.

Sketch: It suffices to check that any complex monic polynomial has a root. Any such polynomial is the characteristic polynomial of some matrix (one can use the so called companion matrix). Thus one is reduced to showing that an nxn complex matrix A has an eigenvalue or equivalently an eigenvector. A may be assumed to be invertible, since otherwise 0 is an eigenvalue. Then A acts on complex projective space $P = \mathbb{C}\mathbb{P}^{n-1}$ by sending the span of v to the span of Av. An eigenvector corresponds to a fixed point under this action. Since the general linear group of $\mathbb{C}$ is connected, A can be connected by a path to the identity I (this can be done explicitly by writing A as a product of elementary matrices and deforming these to I in the obvious way). It follows that the A is homotopic to I, and therefore its Lefschetz number on P coincides with the Euler characteristic of P which is nonzero. Therefore, A has a fixed point on P.

Added: I should probably have pointed that the conclusion follows from the Lefschetz fixed point theorem, which was the sledgehammer that I was alluding to.

Sketch: It suffices to check that any complex monic polynomial has a root. Any such polynomial is the characteristic polynomial of some matrix (one can use the so called companion matrix). Thus one is reduced to showing that an nxn complex matrix A has an eigenvalue or equivalently an eigenvector. A may be assumed to be invertible, since otherwise 0 is an eigenvalue. Then A acts on complex projective space $P = \mathbb{C}\mathbb{P}^{n-1}$ by sending the span of v to the span of Av. An eigenvector corresponds to a fixed point under this action. Since the general linear group of $\mathbb{C}$ is connected, A can be connected by a path to the identity I (this can be done explicitly by writing A as a product of elementary matrices and deforming these to I in the obvious way). It follows that the A is homotopic to I, and therefore its Lefschetz number on P coincides with the Euler characteristic of P which is nonzero. Therefore, A has a fixed point on P.