I see that this question was last answered over 4 years ago, so I would like to provide a more optimistic answer that is up-to-date.
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?
Augmented Reality headsets are the type of headset that displays an overlay on a transparent screen. VR headsets will obstruct your view to ensure maximum immersion, so they are not ideal for most people to use with typing.
Microsoft's HoloLens is a cutting edge AR headset that may be what you're looking for. However, it's not really a consumer friendly product yet. ($3500 USD!)
With Microsoft behind HoloLens, it has a pretty good chance of becoming a standardized AR experience for Windows. At some point in the future an iteration of the HoloLens may be a viable alternative to traditional computer screens.
A little less expensive is the Magic Leap One at $2295 USD. This system has its own computer built-in, so it is wireless in a sense. It will be interesting to see how this device will be put to use by the companies currently developing for it.
These AR headsets are still in the early adoption phase though, where developers and companies are still experimenting and finding uses for them. They are not intended for consumer use and the software to enable using them with a desktop environment may not be fully functional. (Or even exist yet.)
If you are OK with VR headsets that obstruct your view of the keyboard, you could use basically any VR headset. There is already software like Virtual Desktop for most VR headsets. There is also a VR desktop environment for Linux.
What kind of virtual resolution would I end up with?
That's a tricky thing to answer. The resolution of the headset ultimately determines the sharpness of text or the detail of objects in 3D virtual space, because virtual objects must be rasterized onto the pixels of that display. The "desktop space" you can use is practically unlimited, considering that you would be free to arrange and resize virtual windows however you want. The actual resolution of these virtual windows that you would really be seeing varies, and it would depend on multiple factors:
- The resolution of the texture mapped to the virtual surface
- The position & orientation of the virtual surface from your viewpoint
- The resolution of the HMD
You could be looking at a virtual screen with 16,000 x 9,000 pixels, scaled to the size of your thumbnail and placed at arms length. Even though there's a very high resolution in the virtual space, it needs to be projected onto your HMD display, and at that distance and size it may be 160x90 pixels on the display. Which would make it impossible to see the detail in that massive texture.
I realize that it would probably be unpleasant to turn around and use desktop space behind me, but I would at least be able to maximize the use of my field of vision
If you did have a 360° panorama of virtual desktop, you could just let the software rotate it instead of turning your head.