Let $f:\mathbb{C}^n\rightarrow \mathbb{R}^m$ be a $\mathbb{R}$-linear map. I want to show that the map $F:\mathbb{C}^n\rightarrow \mathbb{C}^m$ $x\mapsto F(x):=f(x)-if(ix)$ is $\mathbb{C}$-linear.

Since $f$ is $\mathbb{R}$-linear we have that $f(z_1+z_2)=f(z_1)+f(z_2), \forall z_1, z_2\in \mathbb{C}$ and $f(\lambda z)=\lambda f(z), \forall \lambda \in \mathbb{R}, z\in \mathbb{C}$.

To show that $F$ is $\mathbb{C}$-linear, we have to show that $F(z_1+z_2)=F(z_1)+F(z_2), \forall z_1, z_2\in \mathbb{C}$ and $F(\lambda z)=\lambda F(z), \forall \lambda \in \mathbb{C}, z\in \mathbb{C}$, right?