Hey Mike!
Nice video 
I've recently built a similar 4x4x4 cube. I'm having some trouble with ghosting on the cube (it seems that the cathodes are not turning off quickly enough). The way it manifests is that when I turn on a pair of a new anode plane and a new cathode column, the LED which at the intersection of the new plane and the old column, lights up for a brief moment. This results in ghosting. It is most likely a hardware problem, but I want to make sure that it's not a software problem. Have you had any kind of similar ghosting problem? Would it be possible for me to see your code, maybe I'll get some insights about what I might be doing wrong...
Thanks,
Andras