A third dimension is going to require large amounts of computing power, luckily enough thats going to happen so I'll start by telling you how your computer is about to get faster - a lot faster.
Architectural advances in microprocessors seem to have slowed down in recent years, indeed what on the market today is much more advanced than mid 90s designed Alpha 21264?. Instead we are starting to see the addition of additional cores on the same chip and special purpose elements being added. All the major CPU companies are now committed to adding multiple cores on a single die and this along with the additional cache memory made possible by smaller geometries will bring us large rises in computing power.
But CPUs are not the only thing getting more powerful. Graphics Processing Units (GPUs) get more performance at a much higher rate than CPUs but have to date always been limited to producing graphics. These days modern GPUs have programmable vertex and pixel shaders and these are starting to be used for general purpose computations in research and even in some applications. With the next generation shaders they will become not only more powerful but also more general purpose. Just as vector units (e.g. SSE, Altivec) have boosted computing power expect GPU shaders to be utilised in a similar way to provide potentially massive performance boosts - in the order of 1000%.
In part 3 I predicted office and casual computer users will turn away from traditional desktop systems. Desktop PCs will still remain, the numbers may be smaller but their computing power will be massively higher than anything available today. When combined with 3D visual displays all this power will transform what our computers are capable of and how they will be used.
Visual Interfaces are just starting
Some people believe the command line is king and all the graphical stuff is superfluous. The command line is just one way of interfacing with a computer and as such it has it's own strengths and weaknesses, it offers a great deal of power but at the cost of complexity and having to remember strange commands and their syntax. The GUI has other strengths and weaknesses, features are more obvious so remembering how things work is not difficult or not required at all, this is much better for the beginner. The power of a command line is difficult to present on a GUI so their simplicity can be a drawback, there is no one "perfect" GUI, each interface method has it's own place.
A third interface method is on the way, it exists in research labs and in select, specialist areas of industry, the 3D GUI is coming.
The human brain has a very large portion devoted to the processing of visual information. Some regard the human ability of abstract thought to be our greatest strength but when you combine this ability with visualisation it becomes a great deal more potent. If you could visualise an equation wouldn't mathematics become easier? Wouldn't the progress of science become faster if when we manipulated equations we could see the results visually in real time? Albert Einstein could do this in his head, If everyone had the tools to do the same, think of the progress that could be made.
I for one think exactly those sorts of tools are coming. I think the future will become more visual, I expect the type of interface in shown in Minority Report will actually appear and will be useful for many tasks. I don't know if we'll do physics this way but it would certainly have interesting results if we could.
The implementation of the interface in Minority Report was somewhat clumsy with all sorts of weird hand movements required and moving could accidentally send files flying off all over the place. I don't expect to see an exact replication of this system but something similar could emerge.
Alternatively, it could be something better: Take a large screen and view it through LCD shutters mounted on the screen, have cameras monitor your eye and hand movements with tactile feedback through gloves so you can feel what you can see. I don't know what such a system will be used for but I bet playing Doom 5 will be awesome. Does this sounds like fantastic futuristic technology which will never appear? Think again, this system already exists .
Already more advanced techniques are in development such as Holographic screens which give even better displays. Of course Sharp are already starting to sell 3D display screens today.
The 3D GUI
When I say the 3D GUI is coming I mean a GUI displayed in three dimensions, not a 2D representation of a 3D space such as games deliver. I mean a real 3D display where objects on screen have Height, Width and Depth. Our displays are one way or another going to gain another dimension, 2D displays are going to seem somewhat quaint by comparison.
There are problems though, one major know problem with real 3D displays is that they can confuse the viewers eyes no end. The human visual system expects to point both eyes at a single point and focus them there. 3D displays create images in front of or behind the screen but don't focus them there, to remain in focus your eyes must focus on the screen yet point at an image elsewhere. The human visual system is not designed to do this and it is the major reason why Virtual Reality has never caught on. Using a large screen (say 60 inch) will mitigate the need for this as it will be further away and depth of field  means images not too far off the screen will remain in focus.
The key to this will be subtlety, making sure the images are not projected too far in front or behind the screen. Your brain may be able to get used to this without causing too many problems. Another technique is make the displays bright, your eyes' irises will shrink and the resulting improved depth of field should allow images to move further away from the screen.
Compute encounters of the third kind
Once we all start getting these 3D screens we will also get applications for them and I expect just as there was for 2D, there will be a lot of experimentation in new kinds of user interfaces and new ways to use them.
There are some pretty obvious candidates for 3D displays, 3D modelling of course stands out (pun intended) as an obvious candidate, as of course are 3D games - where monsters will really try and bite your head off. I will leave the reader to probe the possibilities of porn.