To keep all information, you also need to store the timezone that the event originally was.
E.g. When comparing temperature charts from different places in different timezones, you may want to show them in a way that "noon local time" is on the same place of the X-axis in both cases.
Or more usual: I keep a log of what I do during a day. If I now move to another timezone and want to look at the log, I don't want the times shown in my current timezone. I want the timezone I was in back then.