To those familiar to playing with higher dimensions, this may appear too restrictive to be of interest. Yet physics on both the quantum and cosmological scales is confined to 3-spatial dimensions. Note that the infinite Hilbert spaces in quantum mechanics a function of the principle quantum number n, not the spatial dimensions. A quaternion equation must generate the same collection of tensors on both sides. The axial vector is the one that flips signs if the order is reversed.
Terms can continue to get more complicated.
This is called a pseudo-scalar, because it does not change under a rotation, but it will change signs under a reflection, due to the cross product. You can convince yourself of this by noting that the cross product involves the sine of an angle and the dot product involves the cosine of an angle. Neither of these will change under a rotation, and an even function times an odd function is odd. If the order of quaternion triple product is changed, this scalar will change signs for at each step in the permutation. It has been my experience that any tensor in physics can be expressed using quaternions.