People would only be confused until those used to the current system (and unable/unwilling to learn a new one) died off.
All we need is a 24-hour clock, and anyone giving a time to anyone else would simply note their zone (0-23). Determine the difference in the two zones, and the larger zone subtracts that many hours to figure out the relative time of day at the other location (and vice versa). There are other ways to calculate the relative difference in daylight/working time between locations too. Some are easier for certain people than others, and some are easier depending on whether or not the difference is =12.
Both systems obviously have pros and cons. The current one means people don't have to calculate anything to know whether it's day or night when already talking to someone. Coordinating action, especially if more than two locations are involved, can take significant calculation though. Changing to a unified time system makes coordination a snap, at the expense of immediately being able to tell the relative daylight/working time in another location.