The issue of how to handle timezone changes is a long one. So far I haven't found a system that does this well without annoying the users. Think carefully what information is needed to correctly do appointments.

Also let's say and have GMT-5 and GMT-6 timezones (TZ). It's absolutely proper for the appointment I entered in GMT-5 to shift an hour when I go to GMT-6 zoned machines. That's no bug!
Bob