With enough engineering (read, money), all sorts of things would be possible, but if you picture the aircraft in a hard, level turn, the g-forces would be pressing "down" relative to a conventional cockpit, but "sideways" for a cockpit that was gimbaled so the pilot was sitting upright relative to the horizon. And that would probably be a lot harder to take than the usual "down" G-forces. Resisting 9 G's pulling your hand and forearm straight down can be done with a support for your arm (side-stick controller) or a bit of a ledge at the bottom of the handgrip (side-stick or center stick), but keeping precise control of the aircraft with 9 G's pulling your hand sideways might be a lot harder.
Beyond that, the sideways G force would be pushing your head to the side, which probably ends up banging your helmet -- pretty hard sometimes -- against whatever restraint or canopy it meets.
Something that achieves some of the goals you mention is the "Virtual Reality" helmet of the F-35, where the pilot can essentially look through the airframe and "see" aircraft, targets, and whatnot even in places that he wouldn't have an actual line-of sight. That allows him to look in any direction, reacting to threats, tracking his target, and so forth. And without having to gimbal the cockpit. Although for what the F-35's cost, that might actually be cheaper!