This article proposes a Bayesian mixed effects zero inflated discrete Weibull (ZIDW) regression model for zero inflated and highly skewed longitudinal count data, as an alternative to mixed effects regression models that are based on the negative binomial, zero inflated negative binomial, and conventional discrete Weibull (DW) distributions. The mixed effects ZIDW regression model is an extension of a recently introduced model based on the DW distribution and uses the log-link function to specify the relationship between the linear predictors and the median counts. The ZIDW approach offers a more robust characteristic of central tendency, compared to the mean count, when there is skewness in the data. A matrix generalized half-t (MGH-t) prior distribution is specified for the random effects covariance matrix as an alternative to the widely used Wishart prior distribution. The methodology is applied to a longitudinal dataset from an epilepsy clinical trial. In a data contamination simulation study, we show that the mixed effect ZIDW regression model is more robust than the competing mixed effects regression models when the data contain excess zeros or outliers. The performance of the ZIDW regression model is also assessed in a simulation study under the specification of, respectively, the MGH-t and Wishart prior distributions for the random effects covariance matrix. It turns out that the highest posterior density intervals under the MGH-t prior for the fixed effects maintain nominal coverage when the true variability between random slopes over time is small, whereas those under the Wishart prior are generally conservative.