You'd stick with your same salary and work time zone.
My client base is America, mainly EST. I wouldn't mind working remotely in the Bahamas. I'd still stay in the EST time zone and I'd just chillax at a resort while working. I'd love to see some kinda long term startup resort/communal workspace. Probably spend 3 months at a time there and fly back to NY to visit friends and family. Definitely spend winter in the tropics.