Looking into the question, it looks like the author accepted an answer 25 minutes after the bounty's grace period ended.
As such, as your answer was the highest-scored answer posted within the bounty period (with a score of at least 2), your answer was automatically awarded half the bounty. (If the author had accepted their answer before the grace period ended, they would have been automatically awarded the full bounty.)
Keep in mind that the bounty owner is notified multiple times about the ending of their bounty, and the expectation is on them that they will adequately award it on time if they want it to go to a specific answer. The provision that half the bounty is awarded to a top-voted answer mostly exists in case the bounty owner intentionally decides not to give the full amount to an existing answer; it helps reward those who made an attempt to answer the question for their effort. So there's nothing "dishonest" in keeping the bounty for yourself; you were awarded half the bounty for your effort, the bounty owner didn't award the bounty despite being notified multiple times of its ending, and the community thought your answer was relevant.
If you really want to hand the bounty over to the other answer, the best thing you can do is to start your own bounty on the question and award it to their answer. Note that unless you're willing to give away some of your own rep, they will also only pocket half the bounty. If the author is really intent on giving them the full bounty, they can try and get a moderator to clear away their previous bounty, so that they can start a new one and give it to the accepted author. Keep in mind that if a moderator chooses to do this, they are doing so out of niceness, and are under no obligation to do such a thing. It's probably easier to just award them the half of the bounty you earned yourself.