Are all vector bundles on a given vector bundle the pull back of a vector bundle on the base? 


In more detail: let $X$ be a space and $p:E\rightarrow X$ a vector bundle over $X$. Let $\iota: X \rightarrow E$ denote the inclusion given by the zero section. Then for $V$ a vector bundle over $E$, we have the restriction $\iota^\* V$ a vector bundle over $X$, while for $W$ vector bundle over $X$ we can consider the pull back $p^\*V$ to $E$. On one hand, it is straightforward that $\iota^\* p^\* W = W$. 


When is it true that $p^\* \iota^\* V$ is isomorphic to $V$? (so that any vector bundle on $E$ is the pull back of a vector bundle on the base)


Clearly the question depends on which setting we are in. 


For $X$ a topological space and $E$ a topological vector bundle this is clearly true, since $\iota\circ p$ is homotopic to the identity in $E$. This can be shown using the classifying space for vector bundles and the fact that homotopic maps give the same vector bundle.


For $X$ a manifold and $E$ a smooth vector bundle this is true as well, using the fact that they are isomorphic at the topological level.


What happens in the holomorphic and algebraic setting? 
Say $X$ a complex manifold, $E$ a holomorphic vector bundle and considering the vector bundles $V$ and $p^\*\iota^\* V$ up to equivalence of holomorphic vector bundles. Or $X$ a smooth algebraic variety (over $\mathbb{C}$ if needed), $E$ and algebraic vector bundle and equivalence of algebraic vector bundles.


I thought it had to be true, but then I realized that for $X$ equal to a point (in the algebraic setting) this is Quillen-Suslin theorem, so that one need to be more cautious...


Another remark: for $V$ of rank one this is true in both complex and algebraic setting, since there is an isomorphism between the Chow groups of $X$ and $E$. 



Vector bundles on vector bundles
This is definitely false in the algebraic or holomorphic setting, even in dimension 1. There is a well-known example (see [this post](https://mathoverflow.net/questions/159058/non-separatedness-of-moduli-space-of-sheaves)) of a rank 2 vector bundle $E$ on $\mathbb{C}\times \mathbb{P}^1$ such that $E\_{|\{t\}\times \mathbb{P}^1 }=\mathcal{O}\_{\mathbb{P}^1}^2$ for $t\neq 0$, but
$E\_{|\{0\}\times \mathbb{P}^1}=\mathcal{O}\_{\mathbb{P}^1}(-1)\oplus \mathcal{O}\_{\mathbb{P}^1}(1)$. Such a bundle cannot be the pull back of a bundle on $\mathbb{P}^1$.


