If you agree that complex numbers are more "complicated" than real numbers, and quaternions are more "complicated" then the complex numbers. What currently known is the most complicated algebra?
I know of one candidate which is the Monster Vertex Algebra.
Why I say that's the most "complicated" is that it's automorphism group is the Monster group, the biggest known sporadic finite group.
So I can't imagine a way for an algebra to be have more complexity than that.
Are there any known algebras that have more structure or complexity than that?
I would define complexity as the algebra cannot be in the middle of some family of algebras in an obvious way. It cannot be easily decomposed into product of smaller algebras.
It presumably would have to be related to something beyond group theory. Perhaps it would have to include some concept of supersymmetry?
Edit: A concrete definition of complexity could be this: From some mathematical logical axioms such as set theory. The complexity of the algebra is the minimum number of symbols that are needed in order to define it.
Your question makes no sense, and I'll try to explain why. The problem is that "complicated" has no concrete definition.
You say that the complex numbers are more complicated that the real numbers. I say they are simpler. It depends on point of view. I understand that you think of them as more complicated because they are constructed from the reals, and in a sense they have more structure. On the other hand, if you try to factor a polynomial over the real numbers, or get the Jordan form of a matrix, you need to consider special cases that simply disappear on $\mathbb C$. Also, the theory of differentiation and integration of functions of a complex variable is a lot nicer than its real counterpart. From the point of view of cardinality, they have the same cardinality so they are equally "complicated".
It happens very often in mathematics that a bigger object is easier to understand than smaller objects it contains. Here are (very) few examples:
It is way easier to understand the whole set of natural numbers than the subset of the prime numbers.
On generalizations of $\mathbb C$, we have the $B(H)$, the set of all bounded linear operators on a Hilbert space. It is a particular example of both a C$^*$-algebra and a von Neumann algebra; and all C$^*$-algebras and von Neumann algebras are contained in $B(H)$ (with some small quirks that I'll avoid mentioning to keep it simple). Still, it is insanely easier to say things about $B(H)$ as an algebra than it is about its subalgebras.
Another generalization of $\mathbb C$ (and a particular example of $B(H)$, when $H$ is finite-dimensional) is $M_2(\mathbb C)$, the algebra of $2\times2$ complex matrices. No one feels threatened by $M_2(\mathbb C)$, yet there are many unsolved questions about its subspaces.