Given a partially computable function, is there an analytic complex function which is equal to it at every point of it's domain? Or under what condition does a partially computable function correspond to the restriction of an analytic complex function?

Any complexvalued function on $\mathbb{N}$ can be extended to an entire function, so the answer is "yes." This follows from Theorem 15.13 of Rudin's Real and Complex Analysis, which states that for any open set $\Omega$ in $\mathbb{C}$ and subset $A$ of $\Omega$ without limit points, there exists a holomorphic function on $\Omega$ taking prescribed values at all the points of $A$. 

