Is he a soulless abomination? Or does he have a soul, corrupted by years of murder and worse? Maybe he doesn't have his soul and he's trying to find a replacement.
Frostmourne steals souls. He picked it up, got his soul stolen. Where is it at that time?
I see three possible possessors: the sword itself, the Lich King, and someone in the Burning Legion since they created Frostmourne.
If Frostmourne has his soul then he is still a slave. Perhaps it even took Ner'zhul's soul during the merger. It would seem a bit silly for the latest greatest threat to the world to be run by a sword.
If the Lich King has his soul, then things are confusing. Did he get it back when they merged? Was it as he left it or had Ner'zhul damaged it? I don't imagine that former orc plays nicely with others toys. If Ner'zhul had it, then what was the point of their dramatic merger? He could have done that without Arthas needing to climb all the way up. Merge them and then Arthas can pick up their combined souls when it is convenient. I realize Arthas was in the area anyway due to fighting off Illidan's forces. Maybe they were already merged, or the merger was in progress while he was ascending to the Frozen Throne. Was it not Arthas, but actually Ner'zhul who was hearing the voices? Was that him shuffling through Arthas' memories, seeing the mind of a human for the first time?
Finally there's the chance that someone in the Legion has it (having taken it through Frostmourne). This seems unlikely since Arthas is allied with the Lich King (or was before the merger) and the Lich King is resisting the Legion.
Maybe his soul is lost now and he's only working with the Scourge in order to harvest souls. He needs so many because he thinks he's just that important.
The Secret World: In the Dusty Dark
3 hours ago