Many references itemize what properties one gives up with each successive application of the Cayley-Dickson construction:
Complex numbers - lose self-conjugate identity, but satisfies the fundamental theorem of algebra (and can represent 2D points or vectors)
Quaternions - lose commutativity (but can represent 3D rotation, which isn’t commutative)
Octonions - lose associativity, except for each of aab and abb
Sedenions - lose associativity of aab and abb
John Baez’s “This Week's Finds in Mathematical Physics (Week 59)” http://math.ucr.edu/home/baez/week59.html concludes with a letter by Toby Bartels explaining why. An excerpt:
I will prove below that the 2^n onions are a division algebra
only if the 2^(n-1) onions are associative.
So, the question becomes: why aren't the octonions associative?
Well, I've found a proof that 2^n onions are associative
only if 2^(n-1) onions are commutative.
So, why aren't the quaternions commutative?
Again, I have a proof that 2^n onions are commutative
only if 2^(n-1) onions equal their own conjugates.
So, why don't the complex numbers equal their own conjugates?
I have a proof that 2^n onions do equal their own conjugates,
but it works only if the 2^(n-1) onions are of characteristic 2.
The real numbers are not of characteristic 2,
so the complex numbers don't equal their own conjugates,
so the quaternions aren't commutative,
so the octonions aren't associative,
so the hexadecanions aren't a division algebra.
But they're pretty weird; very few of the rules you're used to for "numbers" apply. Ditto for the further constructions.