The space of continuous functions from a topological space to a complete one is always complete. The analyticity of a function in a closed condition in this topology (a uniform limit of holomorphic functions is holomorphic, and we can work locally if needed) so the space of holomorphic functions from the projective plane to itself is complete. This is precisely $\mathbb C(t)\cup\lbrace\infty\rbrace$, for $\infty$ the constant function equal to $\infty$.

I believe this argument shows that the space of holomorphic functions between two complex manifolds with a smooth metric on the target is complete for the uniform topology if and only if the target is complete.