[reikonomusha]

It doesn't make sense in and of itself. We usually think of division as a closed operation: we divide two things (like real numbers) and we get the same kind of thing out (another real number).

In Hamilton's original view of quaternions, he defined a "geometric quotient" of two 3d directed lines (one kind of object) as being a quaternion (another kind of object), and gave all sorts of complicated geometric formulas for how to calculate it.