Noncommutative algebraic geometry is a developing field. Things have not yet got the final form as in commutative geometry.
But one might wonder whether things are any better in the case of skew-fields, ie division rings, ie possibly noncommutative rings in which each nonzero element has a multiplicative inverse. Algebraic geometry is much simpler in the case of fields. So are things better in the case of skew fields? For instance, is there any particularly nice geometry over the quaternions, like there is one over the complex numbers?