This is completely confusing and off-putting. No-one would know whether to stop, keep typing, wait to see if someone submits a decent answer, or what.
Unless you can somehow stop "bad answers" only, we don't want to put people off answering. Let those answers come through, dupes or not, and let the OP and community decide which is the best via voting.
Answers are not just to answer the OP's issues, they're to provide future ref for others, and so multiple answers are fine as long as they're decent, add something new (even if just a different approach), are correct, and are topicnessly sound.
If not, they don't get voted up, or they get voted down.
Sometimes a subtle difference between similar answers is what made me understand, or one specific approach better suited my scenario.
Good vs Evil
If two good answerers are writing, and one stops as they believe someone else is sorting it, we lose a good answer. If they both stop, as we cannot realistically notify when someone has "stopped" writing, then we lose two (or all) good answers.
Also, it's possible one of the multiple users typing an answer is writing a "poor" quick answer just to get in first to snaffle the rep, and they would certainly plod on and post their crap answer.
Then we end up with less choice to vote up or down.
See Fastest Gun in the West Problem for more on how this can be
helpful for "quick replies" versus "thought out replies".
Quick answers can be good - depends on the question, the answerers typing and thinking speed, etc.
And, your proposal would make FGITW worse.
With "bad" quick answers, where a user slaps in a quick response just to "get there first", when that user suddenly see a message "Someone else is typing an answer" then the heat is on and they're gonna speed up.
So upon seeing the notification and a race, they likely post their answer abruptly, with potentially less info as they don't bother writing the additional info/advice they might otherwise have.