There is an urgent need to incorporate the perspectives of culturally diverse groups into AI developments. We present a novel conceptual framework for research that aims to expand, reimagine, and reground mainstream visions of AI using independent and interdependent cultural models of the self and the environment. Two survey studies support this framework and provide preliminary evidence that people apply their cultural models when imagining their ideal AI. Compared with European American respondents, Chinese respondents viewed it as less important to control AI and more important to connect with AI, and were more likely to prefer AI with capacities to influence. Reflecting both cultural models, findings from African American respondents resembled both European American and Chinese respondents. We discuss study limitations and future directions and highlight the need to develop culturally responsive and relevant AI to serve a broader segment of the world population.