Improve timepicker experience

Allow both selection clock time and duration.

Add dropdown for hours and minutes. By default could be good option to show just quarter hours.

No file, test, or entry point is named, so first locate the timepicker component and read its current selection behavior. Define the interaction and verify that it supports both clock-time and duration selection, hour and minute dropdowns, and an optional quarter-hour default.

Written by the indexing model from the issue text.