One way to think of it is that numbers are mainly used to solve problems by studying the behavior of subsets of them that have specific properties. Like "prime

One way to think of it is that numbers are mainly used to solve problems by studying the behavior of subsets of them that have specific properties. Like "prime numbers", "square-free numbers", and "solutions to this given equation", etc., and the interesting feature of numbers in general is that they can embed multiple such concepts at the same time. This is helpful because you can find numbers that share multiple properties in order to bridge between otherwise unrelated problems -- it makes it easier to compose solutions of simpler problems into solutions for larger, more complex problems. From that point of view, quaternions and octonions solve problems in the same way more common numbers do, they just have different sets of properties and so help solving a different set of problems. Of course, they can only do this if we study them well enough to have a sufficiently large suite of concepts and relations in our toolbox.