Nonintegrable almost complex structures

Question

The Newlander-Nirenberg theorem states that any Integrable Almost Complex manifold is a complex manifold. I am looking for natural examples of almost complex structures that are not integrable.

Answer 1

The sphere $S^6$ naturally lies inside of the imaginary octonions $\operatorname{Im}\mathbb{O}$. At the point $p\in S^6$, multiplication by $p$ on $ T_p S^6 = p^\bot \subseteq \operatorname{Im}\mathbb{O}$ defines an almost complex structure.

This almost complex structure is not integrable, due to the non-associativity of the octonions.