TITLE: Do any significant changes happen in hypercomplex numbers beyond the eight dimensions of the octonions?
QUESTION [5 upvotes]: They continue in the fashion of powers of 2: reals (1), complex (2), quaternion (4), octonions (8), and then there is sedonions(16), right?  And, this keeps going, right?
Do any significant changes happen in hypercomplex numbers beyond the eight dimensions of the octonions, the way octonions mark where associativity is lost?

REPLY [2 votes]: Since the OP indicated he is interested not only in Cayley-Dickson construction, here is an answer about other algebras.
In tessarines or multicomplex numbers (which are isomorphic to each other), nothing special happens after dimension $4$.
At dimension 4, zero divisors will appear, but in all further dimensions equal to powers of $2$ the algebras remain commutative, associative and algebraically closed.
In split (hyperbolic) construction, that skips usual complex numbers, zero divisors appear at dimension $2$. The algebra remains commutative and associative at any power-of-2 dimension, but it is not algebraically closed.