Yes you can, the alternator output will keep the system live, but not a good idea.
This is why the cut out switches used on track cars have to be approved, this is because not only do they disconnect the battery live, they also cut the ecu power(supposed to cut coil but ecu does same job), they also have a small resistor going to earth from the live terminal that grounds when turned off, otherwise it would fry the alternator over time.
Some very new clever systems may cut the engine, when they sense the different resistance between battery and not, but certainly anything cossie age will run merrily with no battery, remember the alternator once rotating and having recieved a positive to get charging under way won't stop outputting till it stops rotating, it's of course connected to the positive terminal leads so that's why it keeps running.
tabetha