Yes you would need a 16V alternator, so you need to make sure your lights and accessories can run at 18V or so while the car is running.

I would be more concerned with why your car may not start well on a 12V system.

Remember you can't buy a 16V battery anywhere for the most part, so if you have a problem on the road your stuck. I work for a battery distributorship and we occasionally get the call of a panicked racer looking for a 16V battery ASAP. 16V is special race car crap, that is not a battery used in any other industry that I am aware of. The panicked racer usually doesn't like the option of running 2 65lb 8 volt golf cart batteries run in series lol.