The Riemann Hypothesis says that all of the non-trivial zeros of the analytic continuation of the Riemann zeta function lie on the critical line z=1/2+iy.
This (only slightly technical) video gives a nice description of the conjecture as well as some of the important consequences with regards especially to the distribution of the prime numbers.
… and here is another! …this one is even better!
Here is a recently calculated contour integral which I find pretty interesting.
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.