You say you would still have timezones, and timezones are approximately the area (all points) of the globe where it is approximately the same time, say all points where it is closer to 8am than to 7am or 9am when the centerline is 8am. With the current system, that makes the edges of time zones perfect north-south meridiens of longitude (which are then tweaked, of course for local reasons.)
Sun rises at approx 6am and sets at 6pm every day on the equator. Go straight north to another point, and in the northern summer, it's sun rises before it does at the equator, and in northern winter it rises after the equator. So, with your system calling sunrise 8am, the edges of a timezone would drift with the seasons and would only be north-south meridiens twice a year at the equinoxes.
If you decided to define the timezones with fixed (non-drifting) boundaries, your system might require a 2-dimensional grid of small timezones with boundaries not just on the east and west but on the north and south, with behaviors relative to their neighbors that would make timing a phone call to another time zone pretty much require computer assistance.
Sun rises at approx 6am and sets at 6pm every day on the equator. Go straight north to another point, and in the northern summer, it's sun rises before it does at the equator, and in northern winter it rises after the equator. So, with your system calling sunrise 8am, the edges of a timezone would drift with the seasons and would only be north-south meridiens twice a year at the equinoxes.
If you decided to define the timezones with fixed (non-drifting) boundaries, your system might require a 2-dimensional grid of small timezones with boundaries not just on the east and west but on the north and south, with behaviors relative to their neighbors that would make timing a phone call to another time zone pretty much require computer assistance.