Let $E\to M$ be a holomorphic vector bundle over a complex manifold. Considering it as a real vector bundle one can complexify $E$ to $E^{\mathbb{C}}$. The bundle $E^{\mathbb{C}}$ is again holomorphic and $E$ is a holomorphic subbundle of $E^{\mathbb{C}}$.
I am wondering what can be said about $E^{\mathbb{C}}$. For example when is $E^{\mathbb{C}}$ biholomorphic to $E\oplus E$? A necessary condition is certainly that $c_1\left(E\right)=0$ since $c_1\left(E^{\mathbb{C}}\right)=2c_1\left(E\right)=0$.
An example would be $\mathcal{O}(k)^{\mathbb{C}}\cong \mathcal{O}(k)\oplus \mathcal{O}(-k)$ on $\mathbb{CP}^1$. In general what can be said about the short exact sequence
$0 \to E \to E^{\mathbb{C}} \to Q\to 0$
and the bundle $Q$? At least $-c_1(Q)=c_1(E)$. When is there the chance of $Q\cong E^*$ and the sequence beening split?