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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 whichthere exists $\A\subset \Q(\beta),$ such that all elements of $\Q(\beta)$ admitat least one eventually periodic representation with digits in $\A$. In thispaper 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 rationalnumber 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 proportionalto 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 awell known result of Schmidt \cite{Schmidt}.

On periodic representations in non-Pisot bases