Cost-effectiveness analyses (CEA) are recommended to include sensitivity analyses which make a range of contextually plausible assumptions about missing data. However, with longitudinal data on, for example, patients' health-related quality of life (HRQoL), the missingness patterns can be complicated because data are often missing both at specific timepoints (interim missingness) and following loss to follow-up. Methods to handle these complex missing data patterns have not been developed for CEA, and must recognize that data may be missing not at random, while accommodating both the correlation between costs and health outcomes and the non-normal distribution of these endpoints. We develop flexible Bayesian longitudinal models that allow the impact of interim missingness and loss to follow-up to be disentangled. This modeling framework enables studies to undertake sensitivity analyses according to various contextually plausible missing data mechanisms, jointly model costs and outcomes using appropriate distributions, and recognize the correlation among these endpoints over time. We exemplify these models in the REFLUX study in which 52% of participants had HRQoL data missing for at least one timepoint over the 5-year follow-up period. We provide guidance for sensitivity analyses and accompanying code to help future studies handle these complex forms of missing data.