Automated medical named entity recognition and normalization are fundamental for constructing knowledge graphs and building QA systems. When it comes to medical text, the annotation demands a foundation of expertise and professionalism. Existing methods utilize active learning to reduce costs in corpus annotation, as well as the multi-task learning strategy to model the correlations between different tasks. However, existing models do not take task-specific features for different tasks and diversity of query samples into account. To address these limitations, this paper proposes a multi-task adversarial active learning model for medical named entity recognition and normalization. In our model, the adversarial learning keeps the effectiveness of multi-task learning module and active learning module. The task discriminator eliminates the influence of irregular task-specific features. And the diversity discriminator exploits the heterogeneity between samples to meet the diversity constraint. The empirical results on two medical benchmarks demonstrate the effectiveness of our model against the existing methods.