Quaternions were made up by Hamilton. They are an extension of complex numbers. It is said that he first introduced "3d tertions". He was thinking what the relation between i and j had to be (in t=a+bi+cj), walking on a bridge in his hometown. On the bridge it struck him: i×j=k (and j×i=-k as holds for all pairs of different bases). So the quaternion was born: q=a+bi+cj+dk.
It didn't get a grip on physics though. Hamilton invented them for the sake of rotation in classical mechanics. Rotation in two dimensions can be described by complex numbers (which are two dimensional). That's why Hamilton thought to extend the complex number to a tertion. A tertion could be represented by a vector in the complex 3D space. It turned out that he needed one more dimension. To incorporate k. This was because although the square of i and j are both -1, the product ij could not be accounted for in a tertion. The multiplication has to be closed. Which requires a k.
Whatever happened to these guys? Are they just mathematical curiosities instead of the heroes they were supposed to be by Hamilton? I read that the whole of special relativity could be defined by them. Only to give rise to a positive metric of Minkowski space. Is that all? Whatever happened to them heroes?