Autoregressive (AR) models are an important tool in the study of time series data. However, the standard AR model only allows for unimodal marginal and conditional densities, and cannot capture conditional heteroscedasticity. Previously, the Gaussian mixture AR (GMAR) model was considered to remedy these shortcomings by using a Gaussian mixture conditional model. We introduce the Laplace mixture (LMAR) model that utilizes a Laplace mixture conditional model, as an alternative to the GMAR model. We characterize the LMAR model and provide conditions for stationarity. An MM (minorization-maximization) algorithm is then proposed for maximum pseudolikelihood (MPL) estimation of an LMAR model. Conditions for asymptotic inference and a rule for model selection for the MPL estimator are considered. An example analysis of data arising from the calcium imaging of a zebrafish brain is performed.