Can a POA move a hospice patient to another state if they set it up before they became incompetent?

I imagine that you could stop hospice, make the move, and re-establish hospice in the new location.

Is the hospice patient really able to be moved? What would be the mode of transportation? How long would the trip be? Is the hospice patient aware of his or her surroundings? Would being moved from a bedroom in one state to a bedroom in another state be meaningful to this person?

NO. Hospice ends at the county line. You do not move a Hospice patient.