Quantile regression has emerged as a useful supplement to ordinary mean regression. Traditional frequentist quantile regression makes very minimal assumptions on the form of the error distribution and thus is able to accommodate nonnormal errors, which are common in many applications. However, inference for these models is challenging, particularly for clustered or censored data. A Bayesian approach enables exact inference and is well suited to incorporate clustered, missing, or censored data. In this paper, we propose a flexible Bayesian quantile regression model. We assume that the error distribution is an infinite mixture of Gaussian densities subject to a stochastic constraint that enables inference on the quantile of interest. This method outperforms the traditional frequentist method under a wide array of simulated data models. We extend the proposed approach to analyze clustered data. Here, we differentiate between and develop conditional and marginal models for clustered data. We apply our methods to analyze a multipatient apnea duration data set.