  Dynamics of the delay rational difference equation
$\displaystyle{z_{n+1}=\frac{\alpha+\beta z_{n-k}}{\gamma - z_{n}}}$ with
complex parameters $\alpha$, $\beta$, $\gamma$ and arbitrary complex initial
conditions is investigated. Existence of prime period two solutions and higher
order periods are ensured in the complex parameters unlike in the case of real
parameters of the same rational difference equation. In addition, a new
dynamical behavior, chaotic solutions of the difference equation are ensured
computationally.
