Details

The directions of the complex numbers and constraint have no simple relation to real directions in space. Just as every point in space can have a temperature, a single quantity, there are some qualities that a point in space can have that require a more complex description than a single number. A quantum wave function (a Pauli 2-spinor) is like this, where every point in space can be represented by two complex numbers under the above constraint. These directions are represented in an abstract four-dimensional space.

For a 2D real rotation on a unit vector, there is a similar constraint. The squares of the lengths of the projections of the vector onto the and axes must sum to 1. The 2D complex rotation has the same constraint except the projections are now the lengths of the two underlying complex components. If either of these lengths change, the other must also to keep the constraint valid. These two complex numbers can also change independently via a change of phase, which does not change their length. This phase change is what gives 2D complex rotation more degrees of freedom than a 2D real rotation. It can exhibit behavior similar to a 2D real rotation, but it can also change its underlying components in more ways than a real rotation.

References

B. A. Schumm, Deep Down Things: The Breathtaking Beauty of Particle Physics, Baltimore, MD: Johns Hopkins University Press, 2004.

G. Arfken, Mathematical Methods for Physicists, 3rd ed., Orlando, FL: Academic Press, 1985.