Cauchy functional equation over the complex field

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 $f(z)=a \bar{z}+b)$?