A severe limitation of the original autoregressive process of order one or AR(1) process is the Gaussian nature of the assumed residual error distribution while the observed sample residual errors tend to be much more skewed and have a much higher kurtosis than is allowed by a normal distribution. Four non-Gaussian noise specifications are considered, namely the normal inverse Gaussian, the skew Student t, the normal Laplace and the reshaped Hermite-Gauss distributions. Besides predictive distributional properties of some of these AR(1) processes, an in-depth analysis of the fitting capabilities of these models is undertaken. For the Swiss consumer price index, it is shown that the AR(1) with normal Laplace (NL) noise has the best goodness-of-fit in a dual sense for four types of estimators. On the one hand the moment estimators of the NL residual error distribution yield the smallest Anderson-Darling, Cramér-von Mises and chi-square statistics, and on the other hand the minimum of these three statistics is also reached by the NL distribution.