Although not without controversy, readmission is entrenched as a hospital quality metric. To-date, statistical analyses for hospital profiling based on readmission hinge on fitting a logistic-Normal generalized linear mixed model. In doing so, however, death as a competing risk is ignored. For clinical conditions with a strong force of mortality, such as a diagnosis of pancreatic cancer, ignoring death can have profound effects. In this work, we build on a Bayesian semi-competing risks analysis framework to propose and develop novel multivariate hospital-level performance measures that jointly accommodate readmission and mortality. We also consider a series of hospital profiling-related goals, including the identification of extreme performers and the bivariate classification of hospitals according to whether they have higher-than-expected or lower-than-expected readmission and mortality rates. Towards achieving these goals, we develop a Bayesian decision-theoretic approach that characterizes hospitals on the basis of minimizing the posterior expected loss for an appropriate loss function.