Hello,

Have you tried just waiting around a couple days? Maybe leaving the cell and returning after three days?

Talk to Wuunferth in the dungeon.

I think I did wait a couple of days, do I also have to leave Windhelm and come back? Did I bug the quest or is there a way to finish it?

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I did but he just wants to teach me destruction magic

I remember I left Windhelm for a long time after arresting Wuunferth then when I returned I was notified that another murder happened. I didn't know at that time that I arrested the wrong one so I didn't bother returning quickly. I don't remember whether it was a rumour I overheard or somebody told me about the murder. But I know I wasn't in Windhelm for a few days.

I too remember coming back and being notified of a murder in the city Then I talked to the mage in the prison And was able to complete the quest however I never did get the necromancy Amulet I'm still holding on to the strange amulet in my inventory.

Just so you're clear on this, but finish BotI isn't the only thing you have to do to get Hjerim. If Ulfric still rules Windhelm, you have to join the Stormcloaks and do a few civil war quests before he lets you buy it. The alternative is to join the Legion, but then you have to do the entire civil war questline until Windhelm is besieged and Ulfric is replaced, and then the new jarl will make it available.

But to finish the quest, leave Windhelm, come back after a few days, talk to Wuunferth.

Wait for the next murder to occur, then go talk with Wuunfurth.