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?
As for hints, a standard thing to try in functional equations is to plug in nice values. For example, $\phi(0)=\phi(0+0)=\phi(0)+\phi(0)$, hence $\phi(0)=0$. Similar steps allow you to conclude $\phi\left(\frac{m}{n}x\right)=\frac{m}{n}\phi(x)$ for any integers $m$ and $n$ and any complex $x$. In other words, your function is at least linear over the rationals.
Such functions do actually exist, but they're really awful. They even exist over the reals. Part of the problem in finding an example is that over the rationals any additive function is also linear. One implication of this is that any continuous additive function is also linear.
