Just bought 2012 328i- four cylinder, automatic and having problems with the start/stop idling

If it's still under warranty, take it to the dealer. It sounds like it could have a vacuum leak.

Ditto. Warranty. But to answer your question, NO, not normal

What do you mean by unsteady start? Does it do it when you first start the vehicle or only when the automatic start stop feature is active?

i agree, not normal, 2012, assuming you didn't carfax, or know the history, what if any warranty was given? are there any check engine lights, or indicators on, i would steer away from a vacuum leak, more a faulty sensor..or better take 5 minutes make a free phone call, to any BMW DEALERSHIP, ask a service writer to take down your VIN, and do a open recall search, to find common or known of errors, such as fuel pumps they had a few years back. let me know on the lights, and do not under any circumstances attempt to repair it yourself....let me know the codes.

The F30 does not have any recalls currently out on the vehicle. The fuel pump recalls BMW had was on one engine called the N54 and was for the high pressure fuel system not the one in the tank. The new 328i use a new twin scroll turbo 4 cylinder called N20. Also the warranty on that vehicle is still active as long as it doesn't have a salvage title 4 year 50,000 miles. The dealer can deactivate this feature is you dont like it so you dont always have to push the button to turn it off every time you restart it.