Let $X$ be a complex manifold and $G$ be a real Lie group acting on $X$ by holomorphic transformations. If $\frak g$ is the Lie algebra of $G$. Suppose $\hat{\mathfrak g}:=\mathfrak g+i\mathfrak g$ and let $\hat G$ be a complex Lie group associated to $\hat{\mathfrak g}$.
What are the situations where $\hat G$ acts on $X$?
First of all, you should assume that $X$ is compact and $G$ is simply connected, otherwise you will have some some serious problems integrating holomorphic vectors fields into a complex group action. Next, assume that your $\hat{\mathfrak g}$ is a complex Lie subalgebra of the Lie algebra ${\mathfrak h}(X)$ of holomorphic vector fields on $X$ (otherwise, there is no way you can succeed). Then, Cauchy-Kovalevskaya theorem (and here you need simply-connected + analytic + compact) plus Lie's 2nd fundamental theorem will tell you that the monomorphism of Lie algebras $$ \hat{\mathfrak g}\to {\mathfrak h}(X) $$ induces a homomorphism $G\to Aut(X)$, the group of biholomorphic automorphisms of $X$. In order to see why simply connected is needed, consider the representation theory of $sl(2, {\mathbb C})$: Only odd-dimensional representations integrate to linear representations of $PSL(2, {\mathbb C})$. In order to see why compactness is needed, consider an open disk in the complex plane and the (commutative) Lie algebra of constant vector fields on it.