Since no one has done it so far let me also mention the Riemann-Roch theorem for complex analytic or algebraic curves. One needs only 1-variable complex analysis to define smooth compact complex curves and rational functions and divisors on them. Now if the degree of a divisor $D$ is $>2g-2$ where $g$ is the genus, then $\dim\mathcal{L}(D)=d-g+1$ where $d=deg(D)$. One does not even need the canonical divisor to state this.
As a consequence one gets many results on meromorphic functions on Riemann surfaces. For instance
there exist nonconstant meromorphic functions.
the Mittag-Leffler problem (find a meromorphic differential form with given poles and given principal parts at the poles) has a solution iff the sum of the residues is 0. There is a similar statement for meromorphic functions that can be proved in a similar way.
any algebraic curve (or Riemann surface) is projective and moreover can be embedded in $\mathbf{P}^3$.