This is just a curosity .I know some proofs of the fact that Every non constant polynomial with complex coefficient has a complex root via using Liouville's theorem in Complex Analysis.Proof goes as follows:
Let $p(z)$ be a nonconstant polynomial, and by contradiction suppose it has no zeroes. Then $f(z) = 1/p(z)$ is an entire function. If P is non constant, $p(z) \to \infty$ as $z \to \infty$, and so $f(z) \to 0$ as $z \to \infty$ and $f$ is bounded.But,then by Liouville's theorem $p$ is constant,contradiction to our assumption.
I have seen one more proof which uses Field extensions and Splitting fields. I would like to know some more proofs because it's interesting to see that something can be proved in many different ways. Do you have any other nice proofs?