For most of what we do on a computer I don't think so though. Text based communication (social media, email, etc), reading books, writing documents, doing accounting, etc. These are all tasks that don't have any intrinsic third dimension, and since the information needs to be projected to a 2d surface regardless (the back of your eyeball) you can't really pack more information in by taking advantage of the third dimension.
I'm sure the third dimension will be used for "modern design" reasons, maybe you'll scroll through planes that come at you from farther away to switch "tabs" (android already basically does that on 2d displays), and we'll probably have some 3d equivalent of gifs, but I think most content really remains a 2d texture on a plane.
Even if I'm wrong, I'm sure the first iterations will be mostly plane based, just because that's what the existing software knows how to render to.