Here is a very beautiful proof of The Fundamental Theorem of Algebra (probably one of the most beautiful theorems in all of mathematics), recently written by myself for an analysis course.

The proof utilizes some very famous machinery from complex analysis including: counting integrals (also demonstrated for the case of polynomials), Cauchy’s integral formula, winding numbers, Jordan’s Lemma, et al. .. and is probably one of the more interesting proofs that I’ve ever seen of the very famous theorem.