Algebraic curves (one-dimensional projective varieties) over the complex numbers are exactly Riemann surfaces. It confuses everyone at first when one is told "curves are surfaces." Almost everyone else calls $\mathbb{C}$ the complex plane, but algebraic geometers call it the complex line.

One can work in any algebraically closed field, say $\mathbb{A}$, the field of algebraic numbers. But analysis only works in $\mathbb{R}$ or $\mathbb{C}$, which are complete.