TITLE: Nonintegrable almost complex structures
QUESTION [8 upvotes]: The Newlander-Nirenberg theorem states that any Integrable Almost Complex manifold is a complex manifold. I am looking for natural examples of complex structures that are not integrable.

REPLY [9 votes]: The sphere $S^6$ naturally sits inside of the imaginary octonians $\operatorname{Im}\mathbb{O}$.  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 nonassociativity of the octonians.