You need 12VDC excitation voltage to the alternator rotor to get the alternator to work .
You may be able to run without a battery, if you had generator, instead of an alternator .
A generator has residual magnetism due to permanent magnets, they have magnetic lines of flux, that when the armature rotates, the lines of magnetic flux, pass through a conductor like copper, will start to produce electrical flow without an external source of power to get things started, like an alternator needs .