Do I need a box spring, or does it work on an adjustable base?

You do not need a traditional box spring. This Sealy Essentials hybrid needs a flat, supportive surface: a solid or closely-slatted platform (slats no more than about 3 inches apart) or a compatible adjustable base, which it supports. A sagging old coil box spring will undermine the pocketed-coil core, so use a sturdy flat foundation instead.