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Its physically possible, since the PT Cruiser uses the same 4-speed automatic transmission as the Neon only with the AutoStick feature added. The problem would be with the Neons PCM...you would proably need the PCM out of the auto PT Cruiser, in which case, you would need to compelely change the enigne and all the wiring as well...
Or you could go and buy a RMVB (reverse manual valve body) which lets you shift the transmission manually...basically turning it into an Autostick.
Then I had another question, does it matter what year you pull the engine and all from the PT Cruiser? Like if I went and yanked one froma 05' would it work in my 01' neon?
No, it wont really matter but you'll need to get the entire wiring harness, that would also include the dash harness as well since Neon electronics wiring isnt compatable with PT Cruiser electrical stuff.