Quaternions are useful for modeling rotations, which become non-commutative in 3+ dimensions. Octonions? They're connected to certain loops (in the abstract algebra sense... they aren't a real group because of non-associativity) for which I have no idea whether there are real-world applications. Group theory gets some play in cryptography, and loops turn out to be useful in survey design.
More screwy are the sedenions (16 basis vectors) which also have zero divisors (i.e. xy = 0 does not imply x = 0 or y = 0).