Complex numbers with conjugate multiplication - field or ...? I could have sworn that when we learned about complex numbers in signals and systems that they form a field in (at least) two ways, depending on multiplication, which is most intuitively described in polar coordinates:
Normal multiplication adds the arguments' phases, while conjugate multiplication subtracts them. 
But, whereas (scalar) phase addition is associative, subtraction is only left associative. 
So what algeraic structure does $\mathbb C$ under complex conjugation form? 
 A: The map $\mathbb C\times \mathbb C\to\mathbb C$ defined by $(z,w)\mapsto z\overline w$ is the standard inner product on $\mathbb C$.   Thought of in this way, it isn't a multiplication in a ring, but rather it is thought of in the same way that an inner product on any other complex vector space $V$ defines a map $V\times V\to \mathbb C$ with certain properties.
But if we think of $\mathbb C$ together with "conjugate multiplication" and ordinary addition as operations on $\mathbb C$, then multiplication is nonassociative (and noncommutative), but still distributive over addition, hence $\mathbb C$ with this structure is a nonassociative ring. It still has some nice properties, like a right (but no left, thanks Hurkyl) multiplicative identity, multiplicative inverses for all nonzero elements, and cancellation.  
I don't know if it falls under a particularly well studied class of nonassociative rings, but there are books written on nonassociative rings and algebras where you might find something.  Like $\mathbb C$ with its ordinary multiplication, the complex conjugation operation $\mathbb C\to\mathbb C$ still has an important role, with its relation to multiplication given by $\overline{z\times w}=\overline w\times \overline z$.
