Here's another complex analysis proof that I heard about for the first time under a week ago (because it was set as a question on a course I am teaching for). Pick a circle large enough for the modulus of p(z) to be greater than |p(0)| everywhere in that circle. Inside that circle take a point w where the modulus of p is minimal (which obviously you can do by compactness). There are many ways of proving that p(w)=0. One can use the minimum modulus theorem (that any point of minimum modulus not on the boundary must be a zero), the open mapping theorem, the local mapping theorem, or an elementary bare-hands argument.