Example of an $\mathbb R$-linear map from $\mathbb C^2$ to $\mathbb C^2$ that is not $\mathbb C$-linear Example of an $\mathbb R$-linear map from $\mathbb C^2$ to $\mathbb C^2$ that is not $\mathbb C$-linear. One class of examples that I can think of is the conjugate linear maps, are there any other importatnt examples?

All you have to do is completely forget about the $\mathbb{C}$ structure, and pick your favorite endomorphism of $\mathbb{R}^4$.  For example, $(a+bi, c+di)\rightarrow (d+ai, b+ci)$ is not $\mathbb{C}$-linear, like most endomorphisms you pick out.
