Let $E$ be a real vector bundle on a smooth manifold $X$. Let $J : E \to E$ be a vector bundle morphism (i.e. $\pi \circ J = \pi$, where $\pi : E \to X$ is the projection map) with $J^2 = -\textrm{id}$. Then $E$ is a complex vector bundle ($E_x$ has a complex structure given by $(a + bi)v = av + bJ_x(v)$). I will call $J$ an almost complex structure on $E$ – for $E = TX$, this is standard.
Is there any relationship between an almost complex structure on a vector bundle $E$ and the existence of a complex structure on $E$ as a manifold? That is, is there anything we can say about $J$ that will help us to determine whether $E$ is a complex manifold?
Another way of viewing this question is the following:
If you have a smooth manifold, which happens to be a complex vector bundle over some other manifold, is it any easier to determine whether or not it has a complex structure?
Best Answer
I think the answer is simply "no." Let's consider a specific case. We will take the base space $X$ to be a complex manifold and $(E, \pi, J)$ to be a (complex) rank $k$ complex vector bundle over $X$. Let us call $\pi: E \longrightarrow X$ a holomorphic vector bundle if there exists a trivialization of $E$ such that the transition functions $$g_{\alpha\beta}: U_\alpha \cap U_\beta \longrightarrow \mathrm{GL}(k, \mathbb{C})$$ of $E$ are holomorphic. One can show that $E$ is a holomorphic vector bundle if and only if there exists a complex manifold structure on the total space $E$ such that the projection $\pi: E \longrightarrow X$ is a holomorphic map. Therefore in the specific case where the base space is a complex manifold and we require that $\pi: E \longrightarrow X$ be holomorphic, we see that the question is equivalent to whether or not a given complex vector bundle over $X$ can be given a trivialization with holomorphic transition functions.
With this point of view developed, I now bring your attention to a relevant discussion on MathOverflow. In particular, the paper
mentioned in the answer the the linked discussion proves that all complex vector bundles over a projective 3-fold are holomorphic. I think you will agree that with the work they need to do there, it is not particularly "easy" to prove that these bundles are holomorphic. Also mentioned in the answer is that whether all complex bundles on $\mathbb{C}P^n$ are holomorphic is an open question for $n \geq 4$. Therefore we can see that at least determining if a complex vector bundle is holomorphic is in general a difficult question.
If we don't require $\pi: E \longrightarrow X$ to be holomorphic, just that $E$ has some complex structure, I'm not sure how much easier this problem becomes.