What time is it? Watches and horology

Although there are "GTM" watches in your budget (most of them Rolex GMT copies) they tend to use the same movement (or clone of it) where the GMT and hour hand is back to front, with no quick set hour hand. So they only really work if you plan on permanently keeping the same two time zones. Omega and Rolex are the cheapest watches with proper GMT movements if you're planning on changing time zones regularly.

Yeah, it’s be nicer to have a quick set hour hand but I wouldn’t say that not having one means the watch doesn’t work - the 24 hour hand on my Steinhart and on my Oris jumps in hour steps, so adjusting it and the hour hand at the same time is a 30s job, really.