Suppose a compact (real) Lie group $K$ acts holomorphically on a complex manifold $M$. Let $G$ be the complexification of $K$. Is there a natural way to obtain an action of $G$ on $M$ extending the action of $K$?
The example of $S^1$ acting on $\Delta \subset \mathbb{C}$ would show that this is not always possible. But this seems pathalogical in the sense that the action does complexify to an action of $\mathbb{C}^*$ on $\mathbb{C}$. Is this sort of behaviour the only obstruction? For example if $M$ is $\mathbb{C}^n$, does the action always complexify?