the construction of the canonical $J$ for a complex manifold is what I'm interested in

Given a complex manifold, you have a bundle of (1,0)-forms within complexified 1-forms which is generated (over $C^\infty$) by differentials of holomorphic functions. This gives a decomposition of 1-forms tensor C into (1,0) and (0,1)-part. Your I is an operator which is equal to $\sqrt 1$ on (1,0)-forms and $\sqrt 1$ on (0,1)-forms. It is in fact real, hence defines a real endomorphism of TM, squared to -Id.

the construction of the canonical $J$ for a complex manifold is what I'm interested in

Given a complex manifold, you have a bundle of (1,0)-forms within complexified 1-forms which is generated (over $C^\infty$) by differentials of holomorphic functions. This gives a decomposition of 1-forms tensor C into (1,0) and (0,1)-part. Your I is an operator which is equal to $\sqrt -1$ on (1,0)-forms and $-\sqrt -1$ on (0,1)-forms. It is in fact real, hence defines a real endomorphism of TM, squared to -Id.

the construction of the canonical $J$ for a complex manifold is what I'm interested in

Given a complex manifold, you have a bundle of (1,0)-forms withing complexified 1-forms which is generated (over $C^\infty$) by differentials of holomorphic functions. This gives a decomposition of 1-forms tensor C into (1,0) and (0,1)-part. Your I is an operator which is equal to $\sqrt 1$ on (1,0)-forms and $\sqrt 1$ on (0,1)-forms. It is in fact real, hence defines a real endomorphism of TM, squared to -Id.

the construction of the canonical $J$ for a complex manifold is what I'm interested in

Given a complex manifold, you have a bundle of (1,0)-forms within complexified 1-forms which is generated (over $C^\infty$) by differentials of holomorphic functions. This gives a decomposition of 1-forms tensor C into (1,0) and (0,1)-part. Your I is an operator which is equal to $\sqrt 1$ on (1,0)-forms and $\sqrt 1$ on (0,1)-forms. It is in fact real, hence defines a real endomorphism of TM, squared to -Id.