An $\mathbb R$-linear map which is also $\mathbb C$-linear

$C^2$ is a $4$-dimensional real vector space with basis $(1,0), (i,0), (0,1), (0,i)$ $T(1,0)C=(1,0)C$ you have $T(i,0)=(a,0)$ $a\in C$, you also have $T(0,i)=(0,b), b\in C$.

You have $T(1,1)=(1,1)$ so $T((1,1)C)=(1,1)C$, so you have $T(i,i)=(c,c)=T(i,0)+T(0,i)=(a,b)$ it results $a=b=c$.

$T(i,1)=T(i,0)+T(0,1)=(a,1)$, $T(-1,i)=T(-1,0)+T(0,i)=(-1,a)$ since $i(i,1)=(-1,i)$ we deduce that there exists a complex number $d$ such that $d(a,1)=(-1,a)$ $da=-1, d=a$ and $a^2=-1$ this implies $a=i$ or $a=-i$.