Countries, companies, and universities are increasingly competing over top-tier artificial intelligence (AI) researchers. Where are these researchers likely to immigrate and what affects their immigration decisions? We conducted a survey $(n = 524)$ of the immigration preferences and motivations of researchers that had papers accepted at one of two prestigious AI conferences: the Conference on Neural Information Processing Systems (NeurIPS) and the International Conference on Machine Learning (ICML). We find that the U.S. is the most popular destination for AI researchers, followed by the U.K., Canada, Switzerland, and France. A country's professional opportunities stood out as the most common factor that influences immigration decisions of AI researchers, followed by lifestyle and culture, the political climate, and personal relations. The destination country's immigration policies were important to just under half of the researchers surveyed, while around a quarter noted current immigration difficulties to be a deciding factor. Visa and immigration difficulties were perceived to be a particular impediment to conducting AI research in the U.S., the U.K., and Canada. Implications of the findings for the future of AI talent policies and governance are discussed.