We study several generalizaions of the AGM continued fraction of Ramanujan inspired by a series of recent articles in which the validity of the AGM relation and the domain of convergence of the continued fraction were determined for certain complex parameters [2, 3, 4]. A study of the AGM continued fraction is equivalent to an analysis of the convergence of certain difference equations and the stability of dynamical systems. Using the matrix analytical tools developed in [4]. we determine the convergence properties of deterministic, and stochastic difference equations and so divergence of their corresponding continued fractions.