The complex numbers are built the real numbers but add an unknown unit which squares to $-1$ (as no number in the reals squares to $-1$) called $i$ for "imaginary", because such numbers were considered a hack at the time of their discovery by mathmetician Bombeli. The sum of a real number and the product of $i$ and a real number is called a complex number, represented in the form $a+bi$. These numbers have found many applications due to the connection between polar complex multiplication and rotation on the 2D plane, as $(r,\theta)*(s,\iota)$ = $(rs,\theta+\iota)$.
Soon after the discovery or invention of the complex numbers, people tried to see if the previously mentioned rotational behavior of complex multiplication could be extended to 3 dimensions. William Hamilton, the person known for creating quaternions, proposed that a tricomplex number represented as $a+bi+cj$ where $i^2=j^2=ij$ would do the trick, but soon he realized he needs a four dimensional system, with 3 dimensions for the rotation, and one for other measures. This system did work, and became known as the quaternions from the Latin prefix quatern- meaning 4.
But the next real-based algebra to find a use after quaternions were octonions, which has 8 dimensions. The next one after that is called the sedenions, and has 16 dimensions, and the next one after that has 32, and so on, with every dimension being a power of 2.A theorem known as the Frobenius theorem proves that these power-of-2 algebras are the only complex-based division algebras, so it is impossible to have a 3-ion or 5-ion number without giving up closure under division.
However another real-based algebra known as the dual numbers might be able to combat this, mainly because they aren't a field/divison algebra to start with. Dual numbers are numbers of the form $a+b\epsilon$, where $\epsilon^{2} = 0 $ and $\epsilon \ne 0$, and have a few applications like automatic differntiation. They aren't a field because $a/b\epsilon$ is undefined, as this would be the equivalent of saying $\sqrt{a/0}$, which explodes to infinity, but they are a ring. Because they aren't a field due to that no-real component case, it is it possible to extend dual numbers to any dimension, and keep them a ring, without violating the Frobenius theorem, as they aren't a field in a niche case where the real components are 0? (For instance a trial or 3-dual number would be $a+b\epsilon+c\zeta$ where $\epsilon^2 = \zeta^2 = \epsilon\zeta = 0$ and $\epsilon \ne \zeta \ne 0$)
