No. Your alternator on a typical car puts out between 12 and 14 amps. This is enough to keep the battery charged and supply power to the vehicle. However, it cannot by itself supply enough power to the vehicle without the battery due to how modern cars now use the battery.

This is why police cars for instance have multiple batteries with larger 24 amp alternators. The alternator alone simply cannot provide enough juice for the vehicle, especially at idle engine rpms.

Also, in modern cars, the battery serves as the ground for the electrical system. Many of the components require the ground in order to not throw codes or function.

While you can potentially run a car without a battery on an alternator, it wouldnt last long or be reliable.

It used to be on older cars, if your batter was dead, you could hook an 'extra' battery up, start the car, and put the battery back in the other car, etc and drive to where you need to go.

You truly can't do that in modern cars. As soon as the battery was disconnected, the car will shut down.