Solving equations over the hyperbolic numbers is weird; for example, the equation $x^2 = 1$ has four solutions; not only are $1$ and $-1$ solutions, but so too are $j$ and $-j$.
Question. Are there any good resources out there for learning how to solve polynomial equations over the hyperbolic numbers? I'm especially interested in the problem of finding all $n$th roots of unity.
Remark. The familiar formula $$x^2 + bx = (x+b/2)^2-b^2/4$$ holds over any ring in which $2$ has a multiplicative inverse. This means that we can reduce the problem of solving $x^2+bx+c = 0$ to the problem of finding the square roots of $b^2/4-c$.