The complex structure of a point in the plane is defined by the linear map

and corresponds to a counterclockwise rotation by . This map satisfies

where is the identity map.

More generally, if is a two-dimensional vector space, a linear map such that is called a complex structure on . If , this collapses to the previous definition.

More things to try:

Weisstein, Eric W. "Complex Structure." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ComplexStructure.html