What is the relationship called between two quaternions that are equal in magnitude, but different in component values? They are not necessarily unit quaternions.

Quaternions are usually normalized to have a magnitude of 1.Then they represent an orientation in 3D space.

Un-normalized quaternions are rarely used in graphics and game work, although they probably have uses in other areas of mathematics.

I use them for fractals.

https://github.com/sjhalayka/julia_n_dimensions_paper_v3/blob/main/julia_n.pdf

Nagle said:
Quaternions are usually normalized to have a magnitude of 1.Then they represent an orientation in 3D space.

A little more on that.

They can represent orientation even when not normalized to a magnitude 1. It is just less helpful for visual details.

They're roughly analogous to polar coordinates or a rotation angle for 2D orientation. In polar coordinates you can rotate as many times as you want around the middle so there are infinitely many matches to the same visual position. We normalize the angle to within a single circle, often either +/-pi or from 0-2pi, so that it will always be a direct representation of the first wrap around the space.

Quaternions can similarly (although in a more complex way) rotate as many times as you want so there are infinitely many matches. If it isn't normalized it could wrap around as many times as you want to reach the same visual position. By normalizing them to magnitude 1, we similarly ensure it's a direct representation of the first wrap around the space.