*Spoilers*

I completed this quest by sending the court wizard to jail. Also, I didn't sell the strange amulet to Calixto so it's now stuck in my inventory (can't sell it or drop it).

After a few days (game time), I return to Windhelm and another murder happens. I go back to the wizard in prison and he tells me another murder will occur later that night around the Stone Quarter. Just when I'm exiting the prison, this last quest appears as "completed" and a new objective "Catch the murderer" is set. Calixto then appears running around the city with the target arrow pointing at him. Problem is I can't kill or talk to him so this quest never ends.

HOPEFULLY THIS WILL BE PATCHED IN THE UPCOMING TITLE UPDATE.

Anyone experiencing this same issue?

I have the same problem and i tried to solve it by not inform the mage in jail and talk to Calixto to see if i were able to sell him the amulet but the option isnt there... i didnt know that amulet was part of the quest by selling it to him, i thought it was just an option if i wanted some gold instead, Everything in the game was working fine until that happened (im level 26 and done both darkbrotherhood and thieves guild quests plus alot of side and misc quest), now it feels like alot of the quest have some glitches/bugs! Like falling through the floor (like noclip mode) ending up in a small cave that i couldnt exit etc etc...and the list goes on from there...I really hope they fix alot of the problems on the next patch!

i have finished both part's of this quest myself and im stuck with the Strange amulet which i never had an option to get identified.

Check the quest log, the game will say patrol the stone quarter at night, basically after 6pm just go to the market area, you will see Calixto behind a woman in an aggressive stance, so either kill him via a weapon or punch him to force him to attack you. That should work. You cant talk to him since it say's "this person is busy" but you really should be able to kill him unless he manage's to get back to that house in which i believe make's it impossible to kill him if i read it right on UESPwiki?

Problem is, the glitch instantly credits you with doing the patrol and you're on "Catch the killer" instantly. Then you can't kill him or anything.