What software and technique would I use to achieve this effect?
Tough question, and I imagine that those effects may be combinations of many other effects (probably in a 3D package given the context of the video itself). Barnaby Roper has a pretty interesting skill set, and we could try forever guessing what he did without getting it exactly right. 

That said, you should be able to accomplish a similar effect with purposefully induced compression artifacts. Getting those artifacts to animate seemingly organically like the video will be tricky.

Otherwise, dig into After Effects and start playing with combinations of all the distortion. Trapcode's Particular is (unsurprisingly) used throughout, but it's hard to pin down exactly what they're using it for everywhere. Still, Trapcode does have a small blog about the video, which might be able to direct you to further expertise:

http://www.trapcode.com/journal/2013/11/19/david-bowie-love-is-lost-hello-steve-reich-mix.html

You may also want to look into Trapcode MIR: http://vimeo.com/46449353; I don't think it was used for the distortion you've highlighted, but it's possible.

Happy hunting!