Go speak to Onmund and he tells you about a trade he did with Enthir. He wants you to go to him and ask him about the amulet. Locate and speak to Enthir. He is HARD to persuade and you can’t buy the amulet from him either. Ask him if there’s any way to get the amulet back.

You now need to get a Staff, the Staff of Charming , which is, as always, in a random location. You need to fast travel to the closest location, get the staff and return it to Enthir.

Return to Enthir once you have the staff and he’ll give you the amulet that Onmund want’s so bad. Now simply go and give it back to Onmund, who will be overjoyed and will join you as a companion anytime you ask now.