(This is a question about the UI rather than the technology required to do it)
What is the clearest way to display a time for events occurring in different timezones
What you have described is a situation where the viewer of the page is supporting the customer.
So I recommend that the design focus on the customer perspective. So, the default, or primary display method should be the local time of the customer. If you are using email or IM, you should have some kind of userdirectory as a source. If you are on the phone, then caller ID can be the basis of time conversion.
This is also an example of how the software is only part of, not the entire support offering. The display should allow easy switching to other time zones, and the phone reps should always be trained to ask what time zone they are in before providing any time-based information.
It should also have an option to show the local time at the point of arrival. This allows you to converse with the customer in a seamless and efficient manner:
CU: "When will it arrive?"
CS: "Based on your phone number, I am assuming you are in EST. The ETA is 8pm".
CU: "I am traveling in Chicago right now. Can you tell me when I should check back?"
(Phone rep taps a "CST" button on the screen, and the displays convert.)
CS: "Sir, if the package does not arrive before 9:10pm, your local time, please call us."