Can there be a notification for those people that participated in the question in the past (either who asked or answered it) once a bounty is started on it?
This would be a good feature to apply.
I mean when a question is featured, it's supposed to be featured and stand out from the rest to attract more attention. So sending one notification to all answerers and even the OP (if the OP is not the user who placed the bounty) to allow them all to have a chance to improve their respective posts for whatever reason which is needed (new updates, newly seen obstacles, etc.).
Hopefully this feature doesn't require a lot out of SE especially with the more developed sites with questions that have tons of answers (for example, Stack Overflow or Programming Puzzles and Code Golf).
This seems like it was a good idea 8 years ago (when the question was posed), 2 years ago (when the question was last answered) and still seems like a good idea today. I wonder if this hasn't happened because it's just low on the priorities or if it's logistically more difficult than I'd expect.
As all other answerers mentioned, this was a must feature when the notification system or the bounty system (whichever came earlier) was implemented.
According to one of the Moderator's answer here, it would overwhelm some people with notifications. But from the vote count of the answer, it is clear that most of the user wants this feature.
If it was implemented at the beginning, folks were adopted it. If it is implemented today, it would be very informative for some people.
For other folks who might become 'overwhelmed', there could be a button in the profile settings to turn on/off the notification for this purpose.