Hi, and welcome to the forum.

Personally, I wouldn't ship a US car to Europe, no matter who actually owns it. But check your loan or lease agreement and you may find a clause in there about what happens if you leave the country. (Not sure about the US, but when I made the move from Germany to France I had to pay up the loan to take my car across the border.)

Anyhow, beside the bank's concerns, there is the little matter of "regularizing" the car to meet European safety standards. In most European countries you have to bring the car up to the local standards in order to register the car in that country - and that can mean replacing the windshield (different kind of safety glass required here), the headlights (used to be the case in France, but I think they changed the requirement - fog lights are subject to strict standards in some countries) and exhaust system.

Depending on the age of the car, some countries may require that you pay VAT to bring the car in - that's 15 - 20% of the value of the car.

You also need to consider what you'll do for parts should you need them. Even "international" car brands produce slightly different versions of the same models to meet local requirements and tastes. A former boss of mine had imported his Nissan sports car (I forget which model) to Germany. He had a fairly minor accident, but the insurance company here insisted on totalling the car because of the cost to get the right parts to fix it.

If you're only going to be over for a year or two, it might be worth it if you can keep your US registration. I think that's usually possible for diplomats and US military types. But under those conditions, you have to re-export the car within 2 or 3 years I think it is - and you cannot sell it during the time it is "temporarily" registered, because they waive paying of the VAT. Check with the US Consulate in Vienna if you're interested in going that route.
Cheers,
Bev