It is well known that the complex plane is algebraically closed: Every polynomial has a zero. The relationship seems, to me, to run deeper: For every complex-differentiable function, there exists a power series which converges to it; power series could be regarded as a sort of a generalized polynomial. Every entire function with a pole at infinity is a polynomial. Etc.
Complex analysis can in some ways be regarded as the successor to Euclidean plane geometry. Many proofs exist of the aforementioned facts, which seem to link the polynomials much more tightly to the complex plane than to the real line, but is there an intuitive, geometric reason as to why these transformations seem so fundamentally linked to what is essentially plane geometry?