I had an unfortunate experience trying to start a bounty.
I chose "Reward existing answer" and typed out a custom message explaining why it deserves a bounty. Wanting to link the answer in the bounty message, I scrolled down to it and clicked "share" to copy the link to it. This closed the bounty pop-up, which meant I lost my message and had to redo starting the bounty.
I'm not sure if this behavior is intended -- it seems only one bounty/share pop-up window can appear at a time. But from a user view, this seems unexpected and likely to cause wasted effort.