Although the new Hamad International Airport at Doha, Qatar, is now in full operation, a hotel alongside the old Doha Airport is still a good option for spotting at both airports.
For the time being, the old Doha Airport still has a few private and GA operators using the runway, and there is at least one stored biz jet here. The new airport is only two miles further away.
The Oryx Rotana hotel is situated alongside the terminal at the old Doha Airport, but according to a recent report if you get a room above the entrance on the top floors, you can see movements at BOTH airports.
Admittedly, the view of the new airport is pretty distant. But with good binoculars and flight tracking equipment or websites, you should see all movements easily, as well as anything still using the old airport.
The Oryx Rotana is part of the Rotana chain of hotels, and is of a good standard as you would expect in the region. You can book rooms through their website, here.