  The dynamics of the second order rational difference equation in the title
with complex parameters and arbitrary complex initial conditions is
investigated. Two associated difference equations are also studied. The
solutions in the complex plane of such equations exhibit many rich and
complicated asymptotic behavior. The analysis of the local stability of these
three difference equations and periodicity have been carried out. We further
exhibit several interesting characteristics of the solutions of this equation,
using computations, which does not arise when we consider the same equation
with positive real parameters and initial conditions. Many interesting
observations led us to pose several open problems and conjectures of paramount
importance regarding chaotic and higher order periodic solutions and global
asymptotic convergence of such difference equations. It is our hope that these
observations of these complex difference equations would certainly be new
add-ons to the present art of research in rational difference equations in
understanding the behavior in the complex domain.
