2004 Isuzu Axiom Shut off

Based on the chain of events, suspect the charging system. The fastest way to tell if it's working is to measure the battery voltage with the engine off, then with it running. With it off, it will be 12.6 volts if the battery is good and fully-charged. With the engine running you must find between 13.75 and 14.75 volts.

No. The battery will only last about half an hour after it has been fully charged. It's the job of the generator and voltage regulator to keep the battery charged while you're driving. You could have a bad battery, but given your description of the symptoms, it sounds like the generator isn't working. It is commonly called the "alternator", but that term is actually copyrighted by Chrysler, the manufacturer that developed it. Everyone will know what you mean when you say "alternator", but to be correct, the industry-standard term is now "generator".

The first half of the diagnosis is done with the two voltage measurements I mentioned previously. If you don't have a digital voltmeter, Harbor Freight Tools has a perfectly fine one for about seven bucks. I can tell you how to use it for these tests. You may need to charge the battery with a home battery charger before taking the measurements. The readings are only accurate when starting with a fully-charged battery.

Slow down. No mechanic would just throw in a new generator without verifying it needs to be replaced first. Did you do those simple tests I mentioned? If you did and you found no change in battery voltage after the engine was started, measure the voltage on the large bolted-on wire on the generator's output terminal. That must always have exactly the same voltage as battery voltage. If it's much higher with the engine running, AND 0 volts with the engine off, there's a break in that wire between the output terminal and the battery positive terminal. Usually there's a very large fuse that is burned open or corroded.

If testing proves you do need to replace the generator, your best bet is to get a copy of the manufacturer's service manual. The closest I can come is to suggest when they're tucked away near the bottom of the engine, they usually come out down there over the cross member. Sometimes you can gain access by removing a wheel or the exhaust pipe.