Pretty simple question. Is there a microcontroller that can output hdmi? If not, what microcontrollers can output any type of video?
It depends on what you are looking for. There are lots o chips made to control and process hdmi frames. However, as Jot said, the arduino is not suitable for this. The arduino can output basic VGA, but it does not have enough "horsepower" to play with HDMI. The best and cheapest solution is to use a Raspberry Pi.