Sounds like a vacuum leak. Does it shift hard too?
The climate control PBU is a sore point these days -- my parts guy got three bad ones in a row, one brand new. Didn't make him very happy, believe me!
Check for hard or broken rubber hoses in the engine compartment, especially on the idle control valve. If leaking, dirt gets into both the engine (very bad) and the idle valve, making it stick -- it goes closed to slow the car, then won't open fast enough when you stop. Most often stalls after an abrupt stop when this is the case. Can be cleaned, but you wil need to replace the rubber hoses, probably all of them.
If you get rough idle ONLY when the brakes are applied, you have a bad brake booster that causes a major vac leak when the brakes are applied only. Happened to me in Canada in the old Dodge Aries. Caught on when I heard a hiss only when I had my foot on the brake.
If it isn't a vac leak, I'm over my head.
1972 220D ?? miles
1988 300E 200,012
1987 300D Turbo killed 9/25/07, 275,000 miles
1985 Volvo 740 GLE Turobodiesel 218,000
1972 280 SE 4.5 165, 000 - It runs!