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    $\begingroup$ For me (and everybody around me) a linear map $J\colon V \to V$ would be called a complex structure. Even if you have a vector bundle $E$, people call an endomorphism $J\in \Gamma(\mathrm{End}(E))$ with $J^2 = -\mathrm{id}_E$ a complex structure on $E$. The "almost" in "almost complex structure" is a complex structure on the tangent bundle that is not integrable. But it makes little sense to ask if a linear map between vector spaces or arbitrary vector bundles is integrable (or you must have a different definition of integrable than the one I know). $\endgroup$
    – Klaus Niederkrüger
    Commented Aug 12, 2019 at 9:00
  • $\begingroup$ I have the impression that there is a bit of confusion here. If you have a complex structure $J$ on the tangent bundle $TM$ you can always find bundle charts for $TM$ such that $J$ looks like the standard complex multiplication in the chosen local bundle trivialization, but the difference between $TM$ and an arbitrary vector bundle is that there are natural (!) trivializations of $TM$ coming from the coordinate charts of $M$. Being integrable means that you can find a coordinate chart of $M$ such that the induced trivialization of $TM$ identifies the fibers with $\mathbb{C}^r$. $\endgroup$
    – Klaus Niederkrüger
    Commented Aug 12, 2019 at 9:12
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    $\begingroup$ @KlausNiederkrüger: I was hesitant to use the phrase 'complex structure' because $V$ is also a manifold, and I didn't want to allude to a complex structure on that manifold (i.e. a collection of charts with biholomorphic transition maps). I have since come across the phrase 'linear complex structure' to describe $J : V \to V$ with $J^2 = -\operatorname{id}_V$ which I think is more appropriate than 'almost complex structure' (again, this could be confused with an almost complex structure on the manifold $V$). $\endgroup$
    – Michael Albanese
    Commented Sep 19, 2019 at 23:43