I've seen it written in a few books and online that an almost complex structure on a manifold $M$ is a complex structure on $TM$.
Could someone explain this to me?
This confuses me and I'm not sure what it means but my impression is the following: All this happens on the corresponding tangent spaces, and locally for $U\subset M$ , $TU = U \times \mathbb R^n$ and so locally , $T(TU) = TU \times \mathbb R^{2n}.$
We get a complex structure on $TU$ by essentially identifying locally $$T(TU) = TU \times \mathbb R^{2n} = TU \times \mathbb C .$$
However, I don't understand what's really going on. Feel like I need to show why the complex structure on TM is integrable.