The univariate alpha power exponential (APE) distribution has several appealing characteristics. It behaves similarly to Weibull, Gamma, and generalized exponential distributions with two parameters. In this paper, we consider different bivariate mixture models starting with two independent univariate APE models, and, in the latter case, starting from two dependent APE models. Several useful structural properties of such a mixture model (under the assumption of two independent APE distribution) are discussed. Bivariate APE (BAPE), in short, modelled under the dependent set up are also discussed in the context of a copula-based construction. Inferential aspects under the classical and under the Bayesian paradigm are considered to estimate the model parameters, and a simulation study is conducted for this purpose. For illustrative purposes, a well-known motor data is re-analyzed to exhibit the flexibility of the proposed bivariate mixture model.