The problem with using quaternions to describe coordinates is that quaternions describe a 4-dimensional coordinate system. This would work okay if you wanted to also include time in your measurement, but that start getting really weird because multiplication with quaternions doesn't work in the way you would think they work which would make calculating distances and changes in time very difficult.
If you are looking for a way to define coordinates for your universe, then the standard coordinate system will actually work just fine, you just have to be very careful how you define it. For example, if you are in a spiral galaxy with a large black hole in the middle, then it would be easy to define regular (cartesian) coordinates with the black hole as (0, 0, 0). We can use the spin of the black hole to define the z-axis, but now we run into a problem: we have no reference point for where to place our x and y axes. This is a problem which is going to come up no matter what coordinate system you use, so you will have to pick some second body as the reference point for the x-axis (once you have the x-axis the y-axis is easy to determine.) This poses its own problems because all of the objects in your universe are going to be rotating on their own accord, so you want to pick a point which will minimize how much coordinates of the other planets in your system change. Note that what we pick as the second point is completely arbitrary because of relativity, so to minimize the pain of a changing coordinate system for other planets, it is useful to pick the other reference point as somewhere in a densely populated area of the galaxy.
Now we have defined a standard coordinate system for the galaxy with the center of the galaxy as (0, 0, 0), and the x, y, and z axes well defined. An alternative system for labeling coordinates would be to use spherical coordinates which are defined by an angle along the xy-plane, an angle between the xy-plane and the z-axis, and a distance. This coordinate system is useful when talking about the orbits of solar systems around the center of the galaxy, but you once again have to be careful where you define your 0 degree angles using the same method as before.
Bonus: Although quaternions aren't used for defining 3D coorindates, they are used for 3d rotations, usually in game programming! See here for more information about them.