Edit: This question has been posted on Math.exchange here. To avoid duplication, please comment on the Math.exchange thread.
I am interested in random complex numbers and am trying to understand why to use complex random variables in statistics?
Complex numbers can equally be viewed as a vector in $\mathbb{R}^2$, but with a defined multiplication. This thread addresses how to do linear regression with complex data.
I am aware complex-valued data arise in areas such as signal processing through complex numbers as a transform of real information. Alternatively, complex numbers can be constructed to represent a pair of real values and provide compact notation in other areas such as economics. A common discussed benefit is compactness of notation, but do we also have calculation or computational benefits by taking advantage of $\mathbb{C}$ and its multiplication?
To ask another way, once you have complex-valued data, are there specific statistical (e.g. distribution, modelling, estimation) benefits by continuing to treat it in the complex field, rather than using real bivariate statistical methods?
