The key fact is that the arithmetic with those number systems implies that they have a special geometry to them (I'll try to say a little about what that means below). But this relationship to geometry constrains what dimensions these number systems can occur in. So 1, 2, 4, and 8 are the only ones where the number systems have the required special properties ("size" of numbers and inverse to multiplication are the crucial properties, but it gets more technical).

(Actually, 8 is a stretch; the octionions don't have associative multiplication, but they almost do. And there is a number system in dimension 16 that's even more of a stretch, but that's really it, for these types of number systems. The corresponding geometry is just too special to happen outside of these dimensions.)

This connection between arithmetic and geometry is really deep, and took mathematicians a loooonng time to figure out. It's not a coincidence that the only special dimensions are powers of two. If you are willing to have fewer special properties on your number system, so the corresponding geometry is less special, then you can have those number systems in other dimensions.

Here's an eli5 version of what I mean about the correspondence between number systems and geometry:

• Real numbers describe forward/backward motion (by addition) and scaling (by multiplication).

• Complex numbers describe rotation (by multiplication); their addition isn't significantly new.

• Quaternion multiplication (dimension 4) describes something more intricate having to do with twists that result from rotation (like when could cords become weirdly tisted).

• Octionions are even more tricky to describe; I don't think I can try here.