I think the accepted answer to your question is "no" - but not because of any Doppler effect. Because your engine is so loud, you couldn't hear anything outside your own a/c.
If you shut your engine off and opened the canopy, the Doppler effect would raise the pitch of the sound of the 109's engine as it closed the distance. After it passed you, the pitch of its engine sound would gradually drop due the the Doppler effect.
In AH its just a cool effect, even if its not true to life.