Characterization of linear operators on $\Bbb{R}^2$ which are $\Bbb{C}$ linear. In my graduation, I got the intersting question in  linear algebra book Hoffmann, Kunge:
Give an example of a linear transformation $T$ on $\Bbb{R}^2$ which is not $\Bbb{C}$ linear, that is, $T$ is not $\Bbb{C}$ linear when $T$ is considered as a map from $\Bbb{C}$ to $\Bbb{C}$.
Then somehow I made this example $T(x,y)=(x+y,x+y)$ as an answer. But now I am curious about how one can define a map $T$ which is both $\mathbb{C}$ and $\Bbb{R}$ linear.
Attempt
If $T:\Bbb{R}^2\rightarrow\Bbb{R}^2$ is $\Bbb{C}$ linear, then $T:\Bbb{C}\rightarrow\Bbb{C}$ is $\Bbb{C}$ linear. Thus there is some $\alpha\in\Bbb{C}$ such that $T(z)=\alpha z$ for all $z\in\Bbb{C}$. Let $\alpha =a+ib$ and $z=x+iy$, then $T$ takes the form $T(x,y)=(ax-by,ay+bx)$ for all $(x,y)\in\Bbb{R}^2$.
So, Is this all the characterisation of such operator? Please correct me if there is any mistake or, give some improvement in the characterisation if possible. Thank you.

Yes, that's right. Another way to say it is that $T: \Bbb{R}^2 \to \Bbb{R}^2$ describes a $\Bbb{C}$-linear map $\Bbb{C} \to \Bbb{C}$ if and only if there exist $a,b \in \Bbb{R}$ such that the matrix representation of $T: \Bbb{R}^2 \to \Bbb{R}^2$ with respect to the "standard" basis $E :=\{e_1, e_2\}$ is
\begin{align}
[T]_E = 
\begin{pmatrix}
a & -b \\
b & a
\end{pmatrix}.
\end{align}
