Do any significant changes happen in hypercomplex numbers beyond the eight dimensions of the octonions?
Yes. The Cayley–Dickson construction doubles the dimension indefinitely, from $\Bbb R$ to $\Bbb C$ to $\Bbb H$ to $\Bbb O$ to the $16$-dimensional sedenions $\Bbb S$ etc. But Hurwitz's theorem tells us $\Bbb O$ is the largest normed division algebra, which somewhat restricts the interest in sedenions. (They include zero divisors, e.g. $(e_3+e_{10})(e_6-e_{15})=0$.) Just as octonions lost associativity but keep alternativity, sedenions lose even this but keep power-associativity, which survives throughout the construction.
The process by which we go $$\mathbb{R}\leadsto\mathbb{C}\leadsto\mathbb{H}\leadsto\mathbb{O}$$ is called the Cayley-Dickson construction. We can keep going more-or-less indefinitely, the next step being the sedenions, $\mathbb{S}$.
- It's also worth noting that there's a lot of flexibility here: we could have also gone from $\mathbb{R}$ to the split-complex numbers instead of to $\mathbb{C}$ if we used $1$ instead of $-1$ in the Cayley-Dickson construction.
However, when we do this things get truly nasty; the obvious horror in $\mathbb{S}$ is the presence of zero divisors, so division breaks down. There are other nastinesses - we have even less associativity in $\mathbb{S}$ than we did in $\mathbb{O}$ (only the latter satisfies alternativity, a weakening of full associativity) - but to my mind that's the most dramatic one.
An interesting question here is how much algebraic nastiness we will ever have to deal with - or, phrased more positively, what are some algebraic tameness properties which the Cayley-Dickson construction will never kill off? I believe there's no good general answer known, but the discussion here will be of interest; for example, we never lose power associativity (basically, that "$x^n$" is well-defined for all $n\in\mathbb{N}$ - this isn't trivial when things aren't associative!).