Seen this, I believe really striking, example yesterday here on MO: although the question https://mathoverflow.net/q/7304/41291 is from 2009, a [post](https://mathoverflow.net/a/37205/41291) by [algori](https://mathoverflow.net/users/2349) suggests that this is still open.

And the really ridiculous conjecture is that *all topological vector bundles on $\mathbb C\mathbf P^n$ are algebraic*.