Why does my 2003 BMW 725 shut off while driving at times?

It appears you may have a bad ignition coil that cuts out after it gets hot. Not uncommon.

You say it shuts off while driving ! or did it shut off an now it won't start again ! Makes a difference , had vehicles that shut off after it got warmed up an then when it cooled down it would start again ! had bad ignition module ! You need to test an see what is missing , spark , fuel . No Start Diagnostic Basics EricTheCarGuy

Check crankshaft sensor.

That can be the symptom of a bad Ignition Control Module, or also a bad crank position sensor. The ICM can be taken off and tested for free at any parts store. The crank sensor circuit can be tested on the car with a multi-meter. Also small motors (like a fuel pump motor) that are starting to fail can sometimes work until they get hot and then stop working until they cool down again.

Next time the car shuts down and won't restart, check for spark on a spark plug wire with the engine cranking over. If you have no spark, suspect the ignition control module or the crank position sensor. If you still have spark, but won't start, check if the fuel pump is putting gas up to the engine.

I'm afraid this is a common problem that has no quick fix. Not common for Mazda cars but common for all brands.
The shop can not fix it if they can't duplicate the problem.
Normally I would say it is electrical and probably a relay failing. Which one is anybody's guess.
If the ignition module or fuel pump fails from heat it takes a few minutes to a couple of hours for the faulty part to cool off and restart.
A small relay can break down and come back to life quickly.
You might try replacing the relays that are most likely to cause the problem but that would only be a gamble.
You might also try looking for Mazda owners groups online and post this question there. A fellow owner might have the same experience.

That would be either a fuel pump quiting or a defective ignition control module.

Generally, the computer only runs the pump long enough to start the engine and build oil pressure. After that, the oil pressure switch keeps the pump on, most of the time by supplying a ground to the fuel pump relay. This is a built in safety thing, when the engine isn't running, the fuel pump shuts off. Most public libraries have Chilton's manuals, or you could pick up a Hayne's manual at a store like Autozone. These books can help you locate the relay and oil pressure switch, as well as provide wire colors for your specific vehicle.

Possibly the coil module under the coil packs gets hot and shorts out because it's on the engine, ( dumb design). I have seen that happen many of times.

sounds like the idle air control motor very common also try to have some one keep it running listen for vacumm leak in back of engine.pcv valve hose sometimes splits.