There's a big list of proofs on MathOverflow [1]. Not a lot of pictures in it but there are many to be drawn.

My favorite is probably top answer: a polynomial f(z) gives you a map from the Riemann sphere ℙ^1 to itself. Topology tells you it's closed and analysis tells you it's open because locally a holomorphic map always looks like z^n. It follows that the map is surjective.

[1] http://mathoverflow.net/questions/10535/ways-to-prove-the-fu...