Given the current hype around artificial intelligence (AI) in health, it is not surprising that we are seeing the well-established role and power of future visions being played out by a range of actors in the area. One example is the surge in the adoption of national AI strategies (NASs). NASs have a structuring function, justifying current political decisions as well as (re)framing the conditions of possibility of AI in society. This paper explores the sociotechnical imaginaries (SIs) of AI in health in the NASs of Denmark, Finland, Norway, and Sweden. The findings reveal that a shared common future vision can be identified, though some variations in the justification of how and why this should be realised exist between the states. This common future vision is of the inevitable and ever-increasing scale and scope of AI in health. This is a process driven and implemented by the private sector, a future which is normalised and legitimised through the promise of greater efficiency and progress. However, a tension between the role of states as simply reacting to, as well as being essential facilitators of, the realisation of this shared vision is identified, a tension that results from technological promises.