The emergence of generative AI, large language models (LLMs), and foundation models is fundamentally reshaping computer science, and visualization and visual analytics are no exception. We present a systematic framework for understanding how human-centered AI (HCAI) can transform the visualization discipline. Our framework maps four key HCAI tool capabilities -- amplify, augment, empower, and enhance -- onto the four phases of visual sensemaking: view, explore, schematize, and report. For each combination, we review existing tools, envision future possibilities, identify challenges and pitfalls, and examine ethical considerations. This design space can serve as an R\&D agenda for both visualization researchers and practitioners to integrate AI into their work as well as understanding how visualization can support HCAI research.