I don’t know if you got the answer you wanted, or if you still need an answer. It is possible to set up calculated date fields for multiple time zones that instantly adjust based on the time zone that you indicate is YOUR time zone. The scenario I believe you were describing would look like the following:
If I select Eastern Standard time as my time zone, then Central will display showing 1 hour earlier, Mountain – 2 hours earlier, and Pacific – 3 hours earlier. If I select Mountain Standard time as MY time zone, then Eastern time displays the time as 2 hrs. later than mine, Central, 1 hr later, and Pacific time 1 hr earlier than my time. This could be done using a number field to specify your time zone to each task, and then using Start1 thru Star4 for the start date / time of each time zone start, and Finish1 thru Finish4 for the finish date for each time zone.
You could have a separate view for each time zone that shows the different times for each time zone, or display all 4 time zones in each task – all dependent on the time Zone that you specifiy is YOUR time zone.
I have an example MS Project file that does exactly this.