Do I need a box spring, or can it go on an adjustable base?

Skip the box spring. This Sealy Posturepedic Plus is adjustable-base compatible, and otherwise it needs a solid or closely-slatted flat foundation with slats no more than about 3 inches apart. An old sagging coil box spring will not properly support the zoned pocketed-coil core and may affect your warranty, so pair it with a sturdy flat base instead.