We address the issue of classifying complex data. We focus on three main sources of complexity, namely, the high dimensionality of the observed data, the dependencies between these observations, and the general nature of the noise model underlying their distribution. We investigate the recent TripletMarkovFields and propose new models in this class designed for such data and in particular allowing very general noise models. In addition, our models can handle the inclusion of a learning step in a consistent way so that they can be used in a supervised framework. One advantage of our models is that whatever the initial complexity of the noise model, parameter estimation can be carried out using state-of-the-art Bayesian clustering techniques under the usual simplifying assumptions. As generative models, they can be seen as an alternative, in the supervised case, to discriminative Conditional Random Fields. Identifiability issues underlying the models in the nonsupervised case are discussed while the models performance is illustrated on simulated and real data, exhibiting the mentioned various sources of complexity.