asiatrails is right, the bandwidth you would need for a high-resolution 360 view would be pretty impractical. If you could do it, you'd probably want it on the helmet- the F-35 can already to that with IR with the DAS, so that would actually be more realistic if you had it manned. Plus, you can look through yourself, so you have absolutely perfect FOV.
Lying down is tempting, but the orientation would be pretty strange in terms of effects of G- getting used to the differences would probably be enough to make even experienced pilots pretty sick. No matter how you orient a person, there's always going to be a G-weak spot, with lying down like you have, you're limited in how fast you can accelerate forward before the pilot's going to Red-out (relative negative G).
That being said, I'm not sure how hard it would be to get used to flying a jet like that- I mean, just flying regularly in the standard position takes a whole different set of reflexes than walking around anyway. So... maybe. It will be interesting to see. For example, I play enough Halo that now I can instantaneously compute moving my right thumb in order to move my "body," so I have a large degree of faith that you can adapt the human mind to almost anything.
Some notes: I think guy lying down would be better off with sidestick controllers, so his hands are at his side, so he's not struggling to keep his arms up in those 13g turns.
I remember a study where test subjects were to wear glasses that turned their vision upside down 24/7. After a few days of misery they started to feel and perform normally. The backside was that when the glasses were removed they saw everything upside down