Sounds like it's thermally (heat) related (thermal expansion of wiring/connections or subsystem boxes). Under the right circumstances a loss of signal/ground or connectivity can be the result. Unfortunately, they can be very difficult to troubleshoot. IMO I'd have a hard time pointing to an alternator as the car restarts once it cools down (if it were an alternator/AC issue the battery should be discharged by now). Crank sensor; maybe. Since it may be a primary suspect you might want to check the connector/wiring for that first and retest (idle until fail/or not).
Next, I'd take a fan and point it at the PCM to see if it is failing due to heat. If it doesn't fail (or extends the time to fail) after a normal heat soak period (30 mins) you're probably in the right area. Alternatively, you could take a heat gun and point it at suspect areas/items to fail a faulty unit/connection early. If you can again you're likely in the right area. Obviously, if using a heat gun be careful Not to melt wiring or plastic components....
The shop should have given you some pointers of areas/items that are suspect if they're following basic troubleshooting procedures. Many shops unfortunately are slaves to their test equipment; if it doesn't tell them exactly what's wrong they're clueless.
For Basic Circuit Checks with No DTC's present check out the PCM Emission Diagnosis manual:
http://iihs.net/fsm/?d=143 Start w/Section 2 - Diagnosis methods and Section 5 - Pin Point Tests for suspect areas you want to check.
In this case sounds like you're losing connectivity of a critical signal (or possibly a ground) once the engine or engine bay reaches a certain temp. If it turns out Not to be the crank sensor you may want to engage in what I term Brute force troubleshooting. Maybe not the most efficient but can be effective.
You can be methodical about it - clean a few connectors/terminals at a time and retest to see if symptoms have changed. When you see a change you're likely on the right track.
I'd go thru the entire harness; starting w/the engine bay PCM connectors & Grounds; disconnect them - check for any loose connectors/corrosion - clean where appropriate and add a little dielectric grease - put them all back together (mark the ones you've checked with a paint marker and retest for symptoms). Take apart the Fuse/BEC blocks - check for any burned/discolored/corroded connectors. If you find any check in the electrical manual (
https://iihs.net/fsm/?d=40 ) to see what circuit/subsystem its a part of which should help you narrow down the problem area.
