For questions about almost complex structures on manifolds or vector spaces.
An almost complex structure on a real vector space $V$ is an endomorphism $J : V \to V$ such that $J\circ J = -\operatorname{id}_V$.
An almost complex structure on a smooth manifold $M$ is a vector bundle endomorphism $J : TM \to TM$ such that $J\circ J = -\operatorname{id}_{TM}$. Note, an almost complex structure on $M$ gives rise to an almost complex structure $J_x : T_xM \to T_xM$ on each tangent space $T_xM$.
Every complex vector space, when identified as a real vector space, has a canonical almost complex structure $J$ given by multiplication of $i$. Similarly, every complex manifold has a canonical almost complex structure. It is a deep theorem of Newlander and Nirenberg that an almost complex structure $J$ on a manifolds comes from a complex manifolds if and only if the Nijenhuis tensor $$\mathcal{N}(X,Y)=[JX,JY]-J[JX,Y]-J[X,JY]-[X,Y]$$ vanishes. See more information here
Related tag: complex-geometry, complex-manifolds, symplectic-linear-algebra, symplectic-geometry
