Suppose that $(E,\pi,M)$ is a complex vector bundle. I've seen it suggested in a few places that if $E$ and $M$ are complex manifolds, and $\pi$ is a holomorphic map, then $(E,\pi,M)$ is in fact a holomorphic vector bundle. In other words, there exist local trivializations which are biholomorphisms. Is this true?
Edit: As explained by Mike Miller in the comments, this cannot be true as stated, because the complex vector bundle might have an "anti-holomorphic" structure to being with. A better question would be whether there is some holomorphic vector bundle structure $(E,\pi',M)$, which is isomorphic as a complex vector bundles to $(E,\pi,M)$. Alternatively, one could drop the requirement that $(E,\pi,M)$ be a complex vector bundle, and instead ask what milder conditions (e.g. $\pi$ is a smooth constant rank submersion) would be sufficient to guarantee that $(E,\pi,M)$ is holomorphic (assuming still that $E$ and $M$ are complex manifolds and $\pi$ is holomorphic).