Cauchy functional equation over the complex field

Question

It is known that the only measurable solutions to the Cauchy functional equation $f(x+y) =f(x)+f(y)$ are the linear ones ($x,y\in \mathbb{R}$). Does the same hold if we take $x,y \in \mathbb{C}$? Edit: After the first answer, I rephrase my question: Are the only measurable functions $f:\mathbb{C} \to \mathbb{C}$ which satisfy the Cauchy functional equation linear or anti-linear (that is of the form $f(z)=a \bar{z}+b)$?

Answer 1

$f(x)=\bar{x}$ satisfies the functional equation, and is not of the form $ax+b$ for any $a,b\in \mathbb{C}$ since $\overline{-x}=-\bar{x} \Rightarrow b=0$ and $ax=x$ for $x \in \mathbb{R}$ would force $a$ to be $1$.

Answer 2

$\mathrm{Re}(x+y) = \mathrm{Re}(x) + \mathrm{Re}(y)$.

and more generally, map $x+iy$ to $u+iv$, using a $2\times 2$ real matrix.