TITLE: In Search of a map $\phi:\mathbf{C}\to\mathbf{C}$ satisfying additivity.
QUESTION [5 upvotes]: I am trying without much luck to come up with an example of a function $\phi:\mathbf{C}\to\mathbf{C}$ such that $\phi(x+y) = \phi(x)+\phi(y),\forall x,y\in\mathbf{C}$ and yet for some $a,\alpha\in\mathbf{C}$ we have $\phi(a\alpha)\neq a\phi(\alpha)$. 
Any hints on how to get me started?

REPLY [5 votes]: What about this ?$$\phi:z \mapsto \overline{z}$$