What do you think happened to Aizen's Zanpakuto after it disappeared?

A zanpakutou is essentially a shinigami's (or Arrancar's) capacity to fight. I think it part of the wielder's soul, even if the personality and strength is on a different level. If he no longer had it, I think it was more than the Hogyoku's will. It seemed more like he really had no need for it, having gained the power to fight without a zanpakutou. I don't think it went anywhere, but really dissapeared, ceased to exist, or however you would like to put it. Just my opinion.

Honestly, I think the Hogyoku basically felt the it was worthless at this point. What's the use in having a blade when you're head and shoulders faster and stronger than anyone else? There's no doubt that some trace remains of it, perhaps strongly enough that he can still use his abilities on those that have seen the release before, but I doubt it. The question is, what has happened since the Hogyoku's power has dissipated? Has the blade returned? It seems unlikely, but then, there's so much we don't know. My guess? The blade's gone, and it's not coming back. I say so sadly, because it's an interesting blade in many ways, and it's a shame to see it go.

I believe Aizen still has his Zanpakuto because he told Ichigo that he had become "one with his Zanpakuto and it's powers"