Work hours, business hours, and sleep hours in most locations will still be tied roughly to sunrise/noon/sunset.
People are still going to want uniform event times within an area--almost no one wants businesses to open 50 seconds later in Seattle than in Redmond, for example.
We will still end up with strips separated by longitude boundaries, with events synchronized within each strip. We'll probably still synchronize each strip so that events are one hour or one half hour later than the strip to their east.
To answer questions like "is it too late to call my friend in <X> now?" where X is someplace outside your strip, you need the offset of your strip from UTC and the offset of your friend's strip. From the difference of those, and the UTC time of normal bed time in your strip, you can figure out if it is probably past bedtime in your friend's strip.
These "strips" are pretty much the same as our current "time zones".
