Consider a second order difference equation in complex plane, \begin{equation} z_{n+1}=\frac{\alpha + \beta z_{n}}{1+z_{n-1}},\qquad n=0,1,\ldots \end{equation}
where the parameters $\alpha, ~\beta$ are complex numbers, and the initial conditions $z_{-1}$ and $z_{0}$ are arbitrary complex numbers.
How to prove that the sequence $\{z_n\}_n$ is bounded?