Detecting positive selection in DNA sequences is important in the study of molecular evolution. The statistical methods available are based on simplifying assumptions, which are known to be violated in real data. The problem is the difficulty in simulating data, which incorporate some of the complexities assumed absent. We explore the use of an under-exploited simulation approach that allows the realization of normally intractable continuous-time Markov chains. The method, jump-chain simulation, can be used to produce data with sophisticated evolutionary models. We use this simulation method to study a context-dependent mutational feature. Referred to as CpG hypermutability, which has potential to mislead statistical models for detecting positive selection. Our analyses suggest that when CpG is present at average intensities, traditional tests are mislead into the presence of positive selection. The work emphasizes the usefulness of jump-chain simulation to improving understanding of the limitations of inference tools, while helping methodological developments.