In the book of Complex Made Simple by Ullrich, at page 4, it is given that
However, if we are considering $\mathbb{C} = \mathbb{R}^2$, then $Df$ is a $2\times 2$ matrix, and not a complex number, but when we consider use the "complex differentiation", $f'(x)$ is a complex numbers, so is the author claiming that
$$T(h) = a * h, \quad a \in \mathbb{C} = \mathbb{R}^2, \quad T: \mathbb{C} \to \mathbb{C} \quad \text{and $T$ is $\mathbb{R}-linear$} $$ for any unit $h \in \mathbb{C}$ ?