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Traditonal Bayesian quantile regression relies on the Aymmetric Laplace Distribution (ALD) due primarily due to its satisfactory empirical and theoretical perforances. However, the ALD dispalys medium tails and it is not suitable for data characterized by strong deviations from the Gaussian hypotesis. In this paper we propose and extension of the ALD Bayesian quantile regression framework to account for fat tails using the Skew Exponential Power distribution (SEP). Linear and Additive model (AM) with penalized splines are used to show the felxibility of the SEP in the Bayesian quantile regression context. Lasso priors are used to acount for the problem of shrinking parameters when the parameters space becames wide. We propose a new adaptive Metropolis-Hastings algorithm in the linear model, and an adaptive Metropolis withing Gibbs one in the AM framework. Empirical evidence of the statistical properties of the model is provided through several examples based on both simulated and real data sets.