You change complex linearity to conjugate linearity, and vice versa, by replacing $I$ by $-I$. It is easier to work in a complex linear coframing; see my lecture notes (https://arxiv.org/abs/1706.09697) where I compute some examples. You avoid this $J$ thing almost entirely.

For question 3, a complex linear map, when realified, can only have an even number of -1 eigenvalues, so a conjugation can't be complex linear on $\mathbb{R}^2$, for example, for any complex structure. On the other hand, in any real dimension which is a multiple of 4 you can clearly have such a complex structure, and there are many.

You change complex linearity to conjugate linearity, and vice versa, by replacing $I$ by $-I$, but only on the domain or the range independently. If you want to change them both, as the same vector space with the same complex structure, it is trickier.

For question 3, a complex linear map, when realified, can only have an even number of -1 eigenvalues, so a conjugation can't be complex linear on $\mathbb{R}^2$, for example, for any complex structure. On the other hand, in any real dimension which is a multiple of 4 you can clearly have such a complex structure, and there are many. If you have a real linear map with simple eigenvalue, to become complex linear, the necessary and sufficient condition is that the real eigenvalues have even multiplicities. You can pick any complex structure on each (even dimensional) real eigenspace, and then pick out any complex eigenvalues in conjugate pairs, making one of them into an $\sqrt{-1}$ eigenspace, and the other into a $-\sqrt{-1}$ eigenspace. For generalized eigenvalues, it is more complicated.

It is easier to work in a complex linear coframing; see my lecture notes (https://arxiv.org/abs/1706.09697) where I compute some examples. You avoid this $J$ thing almost entirely.

You change complex linearity to conjugate linearity, and vice versa, by replacing $I$ by $-I$. It is easier to work in a complex linear coframing; see my lecture notes (https://arxiv.org/abs/1706.09697) where I compute some examples. You avoid this $J$ thing almost entirely.

You change complex linearity to conjugate linearity, and vice versa, by replacing $I$ by $-I$. It is easier to work in a complex linear coframing; see my lecture notes (https://arxiv.org/abs/1706.09697) where I compute some examples. You avoid this $J$ thing almost entirely.

For question 3, a complex linear map, when realified, can only have an even number of -1 eigenvalues, so a conjugation can't be complex linear on $\mathbb{R}^2$, for example, for any complex structure. On the other hand, in any real dimension which is a multiple of 4 you can clearly have such a complex structure, and there are many.