2004 Hyundia Tiburon Battery replacement Jump started my Tiburon but car woun't stay ruuning and died shortly afterward. Is this more than a dead battery?

Sounds like your alternator is bad, if car will stay running long enough to get over to an auto parts place where they can do an alternator test for free have it check or you can remove alternator and take it to them so they can test it. or with a volt meter start car remove jumper cables and check voltage at battery, should have no less then 13.5 volts, what your checking is alternator output voltage.