Alternative Almost Complex Structures

Let us first deal with linear algebra. Assume a matrix $J$ satisfies $J^k= -Id$. Then, there exists a poylnomial $P$ whose coefficients depend on the eigenvalues of your $J$ such that $P(J)$ is a complex structure. Moreover, if your matrix $J$ is a smooth (1,1)-tensor on a manifold then the polynomial is the same at all points of the manifold.

Now, concerning differential geometry, your alternative complex structure $J$ have the same eigenvalues (with multiplicities) at all points. Then, the polynomial $P$ is the same at all points. By [Lemma 6, arXiv:0904.0535], the complex structure $P(J)$ is integrable.

Moreover, if
the Nijenhujs tensor of $J$ vanishes,there exists a coordinate system such that the entries of $J$ are constants by Thompson, G.: The integrability of a field of endomorphisms. Math. Bohem. 127 (2002), no. 4, 605–611.

If the coefficients of your tensor are constant in a certain coordinate system, the Nijenhujs tensor of it vanishes.

Thus, as in the case or ordinary almost complex structure, the vanishing of the Nijenhujs tensor decides about integrability.

Now, about combining your alternative complex structure with another structures such as metric or symplectic structure, the only reasonable formula I can imaging is in [Section 5, arXiv:1103.3877] and it does not give anything very interesting. If you have other ways of combining, please tell us.

Sorry for bad news.

Wait: if we ask for $J^3=−I$, then $J=−I$ is one solution, and that doesn't determine an almost complex structure, so Vladimir's answer doesn't quite cover all possible cases. If $J^k=−I$ and $k$ is even, then clearly Vladimir's answer is ok: $J$ commutes with a unique almost complex structure, and the whole story reduces to the story of that almost complex structure.