  We study periodic expansions in positional number systems with a base
$\beta\in\C,\ |\beta|>1$, and with coefficients in a finite set of digits
$\A\subset\C.$ We are interested in determining those algebraic bases for which
there exists $\A\subset \Q(\beta),$ such that all elements of $\Q(\beta)$ admit
at least one eventually periodic representation with digits in $\A$. In this
paper we prove a general result that guarantees the existence of such an $\A$.
This result implies the existence of such an $\A$ when $\beta$ is a rational
number or an algebraic integer with no conjugates of modulus $1$.
  We also consider eventually periodic representations of elements of
$\Q(\beta)$ for which the maximal power of the representation is proportional
to the absolute value of the represented number, up to some universal constant.
We prove that if every element of $\Q(\beta)$ admits such a representation then
$\beta$ must be a Pisot number or a Salem number. This result generalises a
well known result of Schmidt \cite{Schmidt}.
