It is well-known that when $a$ is any non-zero real number, the most general solution of $f(x+a)=f(x)$ should be $f(x)=\Theta(x)$, where $\Theta(x)$ is an arbitrary periodic function with period $|a|$ .
Now when $a$ is a non-real complex number, what is the most general solution of $f(x+a)=f(x)$ ? I don't believe that $f(x)=c$ only, where $c$ is any complex number.