I've just found a strange amulet in my windhelm home. Can anyone shed any ideas on it???

It may be related to the Blood on the Ice quest

I've completed the quest???

Yes, I think you're correct.

@OP: Have you helped with trying to investigate the murders of young women in Windhelm? Do that quest and the amulet might become relevant.

It's all solved. Do I need it for something else or can I sell it?

I've finished that quest(blood on the ice), killed the killer, got the amulet, and I still don't know what it's for. I plan on enchanting it because it looks SOOO cool with the skull and all.

Isn't that the one that's supposed to turn into the Necromancer's Amulet at some point in the quest line? If so.... I think that's just another one of the bugged bits....

You can pick up the strange amulet in the windhelm murder house, then later you can sell it to the murderer and when you pick it up from him after you kill him it turns into another amulet. But if you hold onto it and dont sell it to him it just remains in your inv as "strange Amulet".

It's unique, man. Hang onto it. If you can offload it, dupe it on a mannequin or something.
Pretend you're totally goth!

If you played the Blood on the Ice Quest in a game started before the 1.4 patch you're stuck with it unless you're on the PC.
If you haven't played the Blood on the Ice Quest then:

I only finished blood on the ice today after getting the update and believing it would be sorted! I went 50 hours in with no issues but now they're around every corner.

I just finished the Blood On The Ice quest tonight and it's an amulet you should find when you're investigating the murders. It'll be in the house you find when following the blood trail. The only way to have it turn into the Necromancer's Amulet is to sell it to the guy who runs that House of Oddities. I found him near the Inn and sold it to him there on the street. I've held back a number of details to not spoil too much.

Yeah, you need to sell it in order to get the transformed version when you finish the quest. The latest patch didnt fix it (unless its intentional).

Just to clarify I finished the quest, bought the house and found the necklace whilst exploring my house.

Yep. I finished part II of the quest last night. Man is this quest bugged. I bought the Windhelm house when Ulfric offered it to me after we won a major battle. Now I found out that the perp runs back to that house after committing his crime even if you own the house, which makes it easy to off him, but then you've got a body inside. However, sometimes the murder doesn't fire. I waited three nights for the perp to kill the woman, and he stood behind the woman for three days and nights. Finally I killed him myself, and then the guards attacked me. So I reloaded, then the murder went off as it was supposed to, and I killed him in sight of the guard and reported the crime solved.

But that stupid Amulet never changed into anything. It's just a 0 weight ugly skull amulet worth 1000 gold that's a plot item that cannot be removed from my inventory and cannot be sold.

And just to double check everything. I unloaded all my gear to see how much weight I was carrying around in plot items. Plot items even after you've completed the quests still don't weigh anything. They just don't stack in your inventory. I was carrying around about 300 lbs in potions.

This quest is bugged for me, I can't get help from the royal what-his-name. Sold the amulet to Calixco, but am stuck at that point . This is all after the latest update too.
Dang.

I even sold the amulet to Calixto, and after the quest got it magically back as "strange amulet" again, and non-droppable. :/ As mysterious as my first Gauldur amulet which apparently suddenly disappeared sometime from my inventory, only to turn up again after a lot of time.

I can't even sell it to calixto......

It's useless. You can wear it. You cannot enchant it. You cannot sell it. You cannot display it.