Retention of skilled workers is essential for labour-intensive organisations like hospitals, where an excessive turnover of doctors and nurses can reduce the quality and quantity of services to patients. In the public sector, where salaries are often not negotiable at individual level, workers increasingly care about the non-pecuniary aspects of their jobs. We empirically investigate the role played by two such aspects, staff engagement and the retention of complementary coworkers, in affecting employee retention within the public hospital sector. We exploit a unique and rich panel dataset based on employee-level payroll and staff survey records from the universe of English NHS hospitals, and estimate dynamic panel data models to deal with the bias due to reverse causality. We find that nurses' retention is positively associated with their engagement, whereas doctors' retention is positively associated with nurses' retention. This heterogeneous response of employee retention can be explained by the hierarchy of workers' professional roles within the organisation.