Unlike Drop D, Drop C lowers every string, so the whole set will feel looser. A heavier set brings back some of that firmness, but it isn't universally required. The right gauge depends on your target feel, scale length, construction and setup.
If you only want a firmer feel on the lowest string, S.I.T. Single Strings let you build that combination. For a full set, Power Wound Nickel is a versatile nickel-plated roundwound, and Power Steel Stainless adds brightness, attack and note definition.
You don't need a separate bass for Drop C, but if a bass will stay in Drop C most of the time, choosing gauges and setup around that tuning gives a more consistent feel. Because all four strings change, check nut-slot fit, relief, action and intonation, and have a technician look at it if you're unsure.