How many "super imaginary" numbers are there? Numbers like $i$? I always wanted to come up with a number like $i$ but it seemed like it was impossible, until I thought about the relation of $i$ and rotation, but what about hyperbolic rotation? Like we have a complex number $$ z = a + bi $$ can describe a matrix $$ \begin{bmatrix} a & -b \\ b & a\end{bmatrix} $$ You can "discover" $i$ by doing (which is used for another discovery) $$ \begin{bmatrix} c & -d \\ d & c\end{bmatrix} \cdot \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ac - bd \\ ad + bc \end{pmatrix} $$ $$ (a + bi) \cdot (c + di) = ac + adi + bci + bdi^2 $$ From here on you can infer that $ i^2 = -1 $.
So what if we do the same thing, but a different matrix? $$ z = a + bh$$ can describe a matrix $$ \begin{bmatrix} a & b \\ b & a\end{bmatrix} $$ and we can discover it the same way $$ \begin{bmatrix} c & d \\ d & c\end{bmatrix} \cdot \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ac + bd \\ ad + bc \end{pmatrix} $$ $$ (a + bh) \cdot (c + dh) = ac + adh + bch + bdh^2 $$ From here we infer that $ h^2 = 1 $.
Also $$ e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots $$ $$ \begin{align} e^{xh} & = 1 + \frac{xh}{1!} + \frac{(xh)^2}{2!} + \frac{(xh)^3}{3!} + \frac{(xh)^4}{4!} + \frac{(xh)^5}{5!} + \cdots \\ & = 1 + \frac{xh}{1!} + \frac{x^2}{2!} + \frac{x^3h}{3!} + \frac{x^4}{4!} + \frac{x^5h}{5!} + \cdots \\ & = \cosh{x} + h \cdot \sinh{x} \end{align} $$
How many more numbers like this are there? And does that mean that for each set of trigonometric functions there exists a number which can turn multiplication into a rotation using those trigonometric functions?
(Sorry if I got some things wrong)