What need's to be fixed is the quest Falion's Secret! =(
But I believe you can buy the house without completion of the quest, but in doing so entering your new home will trigger the quest, I could however be horribly wrong xD It worked fine for me on all my characters but one =/
Seconded - started a new character and now the quest won't start, the first victim just doesn't appear. People in Windhelm are still banging on about it, but no corpse to start off the investigation.
Buying the house http://elderscrolls.wikia.com/wiki/Hjerim in Windhelm will also start this quest.
Another way to start the quest is to steal the key to Hjerim from http://elderscrolls.wikia.com/wiki/Tova_Shatter-Shield and enter the house. When one of the fliers located on a small dresser is taken, the quest will start at step 9.
Buying the house http://elderscrolls.wikia.com/wiki/Hjerim in Windhelm will also start this quest.
Another way to start the quest is to steal the key to Hjerim from http://elderscrolls.wikia.com/wiki/Tova_Shatter-Shield and enter the house. When one of the fliers located on a small dresser is taken, the quest will start at step 9.
Buying the house http://elderscrolls.wikia.com/wiki/Hjerim in Windhelm will also start this quest.
Another way to start the quest is to steal the key to Hjerim from http://elderscrolls.wikia.com/wiki/Tova_Shatter-Shield and enter the house. When one of the fliers located on a small dresser is taken, the quest will start at step 9.