Why aren't Complex number fields used in cryptography? Most of the cryptography is limited to use of real integer fields. Complex numbers are seldom used. Is there any reason for it? Wouldn't complex numbers provide more structure?

If there is already some work on this area, I would be happy to learn about it.
Why aren't complex number fields used in cryptography?


If you mean fields of the form $k[i]/(i^2 + 1)$ where $i^2 + 1$ is irreducible over the base field $k$, well, they are!  FourQ is a twisted Edwards curve $-x^2 + y^2 = 1 + d x^2 y^2$ over the quadratic extension field $\mathbb F_{p^2} = \mathbb F_p[i]/(i^2 + 1)$ of Mersenne prime characteristic $p = 2^{127} - 1$, where $$d = 25317048443780598345676279555970305165i + 4205857648805777768770$$ is nonsquare.  FourQ is appealing because it takes advantage of cheap arithmetic modulo a Mersenne prime on a curve together with a cheap endomorphism.  The coordinate field is in a sense the ‘complex’ extension of $\mathbb F_{2^{127} - 1}$ arising by adjoining a square root of $-1$.

(FourQ is not universally appealing: it has a relatively high cofactor, $8\cdot7^2 = 392$, and it does not provide modern standards of twist security, so it's not a drop-in replacement for X25519 or Ed25519.  But it is implemented in practice by Microsoft's FourQlib and probably deployed somewhere.)