Complete embedded constant mean curvatures (CMC) surfaces in $\R^3$ seem to be highly transcendental objects, and their moduli spaces are generally only understood in a few special cases. In this talks, we will reveal a surprising connection with complex projective structures and holomorphic solutions to Hill's equation $$ U_{zz} + q(z)U = 0 $$ where $q(z)$ is polynomial on $\C$ (really, a holomorphic quadratic differential $Q = q(z)dz^2$, obtained by taking the Schwartzian of the developing map for the projective structure). This allows us to explicitly work out the moduli space of k-ended coplanar CMC surfaces of genus 0 in terms of k-point projective structures on $\C$, that is, those projective structures whose associated curvature-1 metrics have exactly k completion points. The latter is shown to be biholomorphic to the affine space of (monic, normalized) holomorphic quadratic differentials on C with polynomial growth of degree k-2, that is, to $\C^{k-3}$; the former is thus diffeomorphic to $\R^{2k-3}=H^3\timesC^{k-3}$. We'll also discuss some related potential applications, inlcuding an explicit description of minimal surfaces in $S^3$.