but I still want more desktop space
Having worked extensively with virtual reality, I will say without any reservation - this really isn't a good idea.
Is there a head-tracking VR headset (either augmented reality or such that a keyboard can still be seen/used while wearing) that could be used as a 360-degree (or even 180 degree) monitor?
Note these are commonly called HMDs and you might have luck further researching these.
A few significant drawbacks with this idea:
- Pixel density for reading text is poor. The Occulus Rift released next year has a resolution of 1080x1200 (per eye, so effectively that resolution). For most people wanting more than 4 monitors (!) this is going to be infuriatingly small, as you will basically only be able to view a 1080x1200 area at any given time.
- Design purposes are different. Virtual reality hardware is nearly exclusively designed to render/create 3D environments (ie gaming) - not 2D. This is primarily a problem because most are still designed to do this, not be "monitor extensions."
- The ergonomics of this would be awful, since you would be rotating your head constantly. If you don't believe this, hold your head at an 45 degree angle for 5 minutes, it will not feel great.
- ... if it feels ok, add a roughly 1/2 pound (or more) weight on your forehead and repeat. VR HMDs are heavy, which matters more if you use it every day like you intend.
- Eye strain. Put your face ~1 inch from a computer monitor for an hour (or eight). You know how people talk about taking breaks from a normal computer monitor at several feet away? Imagine ~1 inch for that time...
- Windows managers/interaction. This is a hard problem and not a "productionized and solved" problem yet. Is it ok if you can't see your mouse/keyboard? Probably.
- Tracking system. If all that's not bad enough, you need some sort of additional system to identify the orientation of your head, in order to feed data back into your computer, in order to tell the graphics engine what to draw (or what your viewport is if you prefer). This is important because if you are using hardware you likely have to use their custom software - this is incredibly non-trivial to get to work the way you want, in many cases. So if you are going "outside" the designed purpose it is very non trivial (especially since you likely need some sort of specific window manager).
Some more specifics of your questions:
Is it wireless?
Using the Oculus Rift, which can be hacked to do what you describe, it has a total resolution of 1080x1200 for each eye, or 2160x1200. Refresh rate is very important for this sort of thing though, slow refresh rates are very disorienting when your entire world is at a say 5 FPS rate.
To have wireless you are basically streaming better-than-HD video at a veryhigh framerate (probably 20 FPS or higher) or it's incredibly disorienting, especially if/when you move your head and the screen doesn't update immediately. Laggy/slow updates will make you sick or frustrated quickly.
What kind of virtual resolution would I end up with?
If you are willing to write your own windows manager/etc you can probably have whatever virtual resolution you want. This route would not be for the faint of heart :)
still want more desktop space
My last point is about this again.
If you want to use VR/AR for something like this you need to determine whether your primary problems with "only" 4 monitors is visible space (at any given time) or limited tiled space (ie positioning windows on top, in different areas). If you are ok with limiting your visible resolution significantly, for a greater ability to layout windows, then VR/AR in its current form is really not your best bet (given the above disadvantages).
There are also higher resolution HMDs available than the Oculus Rift. But the Oculus Rift has a much larger community of people exploring this idea than other technologies.
My recommendation would overwhelmingly be to get higher resolution monitors and/or more of them if you are looking for an actual, day to day, use case. The VR/HMD route is probably fun and interesting but at this point I highly doubt you would find it useful or beneficial for day to day work.