Depends on what you mean by "really complicated."
If you know Brouwer's fixed point theorem in the plane and do not consider that to be complicated, then no. The curious can DM me and I will share a PDF of this little article (it is three pages).
If you know some basic Algebraic Topology (homology), Hatcher gives a proof in section 2.B for the theorem (actually, he proves something even stronger) in a little under a page.
