We investigate the dynamics of the second-order rational difference equation in the title with complex parameters and arbitrary complex initial conditions. Very little is known about rational difference equation in the complex domain. The solutions of such equation exhibit many rich and complicated asymptotic characters. Analysis of the local stability of the equilibria and periodicity of solutions has been carried out. We further exhibit several interesting characteristics of the solutions of this equation, using computations, which do not arise when we consider the same equation with positive real parameters and initial conditions. We pose several open problems and conjectures of paramount importance regarding boundedness and global asymptotic convergence. It is our belief that the analysis done here is generic in nature and can be helpful in the study of other difference equations in the complex domain.