Model misspecification can compromise valid inference in conventional quantile regression models. To address this issue, we consider two flexible model extensions for high-dimensional data. The first is a Bayesian quantile regression approach with variable selection, which uses a sparse signal shrinkage prior on the high-dimensional regression coefficients. The second extension robustifies conventional parametric quantile regression methods by including observation specific mean shift terms. Since the number of outliers is assumed to be small, the vector of mean shifts is sparse, which again motivates the use of a sparse signal shrinkage prior. Specifically, we exploit the horseshoe+ prior distribution for variable selection and outlier detection in the high-dimensional quantile regression models. Computational complexity is alleviated using fast mean field variational Bayes methods, and we compare results obtained by variational methods with those obtained using Markov chain Monte Carlo (MCMC).