We study alternative models for capturing abrupt structural changes (level shifts) in a times series. The problem is confounded by the presence of transient outliers. We compare the performance of non-Gaussian time-varying parameter models and multiprocess mixture models within a Monte Carlo experimental setup. Our findings suggest that once we incorporate shocks with thick-tailed probability distributions, the superiority of the multiprocess mixture models over the time-varying parameter models, reported in an earlier study, disappears. The behavior of the two models, both in adapting to level shifts and in reacting to transient outliers, is very similar.