The complex vector bundles on complex projective space $\mathbb{CP}^n$ are explicitly classified for low dimensions. When $n\leq 3$, they are exactly the holomorphic vector bundles; when $n\geq 4$ we can use the Schwarzenberger condition together with the result of A. Thomas to classify complex vector bundles according to chern classes.

In general we have the Grassmann manifold $\text{Gr}(\mathbb{C}^{\infty})$ as the classfying space of complex vector bundles, but it is not so useful for computation (e.g. to describe the set of isomorphic classes $\text{Vect}^n(X)$ and topological $K$ group).

Now I wonder whether we have some known examples on the classification of complex vector bundles over other compact complex manfolds e.g. hypersurfaces.