Well, the way to turn rotations and translations into one and the same concept is to embed your 3D space into a higher dimensional space that adds one extra dimension.
The reasons for this are that certain very practical, not at all unusual transforms become much easier that way. They obtain a much more natural description that is easier to work with.
In particular, you'll get screw-transforms.
Another reason is to stop the center point from being a "special point". In nature, in the real world, there is no inherent difference between any two given points in space. All difference simply comes from what particles are in its direct neighborhood.
In a naive mathematical model of space, you get some kind of singularity in the origin of your coordinate system. This singularity is entirely artificial and you can get rid of it in a particular point in space, simply by shifting your coordinate system around. However, naively you can't get rid of it altogether.
To do that, you need to add another dimension. If you do that, you get homogeneous coordinates.
If you do both those things, you end up with a conformal model of space which is as well-behaved as it could be.
It has lots of advantages:
- The Origin is no longer a special point of your space
- You can do all screw transforms in a very direct, simple way, since translation and rotation ends up being the same thing
- Additionally, points (not vectors), spheres and planes become one and the same object for, respectively, 0, finite and infinite radius, which is very natural and easy to deal with.
- Conversely, circles and lines also become one and the same, and a circle of radius 0 becomes something like an infinitesimal vector - a point with direction, which is a great notion for a normal vector.
- Things that actually should be different quantities, like points and vectors, are usually mingled into a single concept which can cause a lot of confusion and head-scratching. In a conformal model of space you get notions that truly separate the two.
- And on top of all that, in a conformal model you actually get rid of the need of a coordinate system. - All your calculations can be done algebraically without ever referring to coordinates, which is a really natural notion.
Coordinates are just a mess to deal with, they add a lot of complexity and they are entirely a mathematical construct with no physical relevance what so ever.
In fact, many misconceptions behind the theories of relativity, and even behind some Newtonian physics stem from failing to see the difference between coordinates and the actual space. So if you can do all your transformations on a space, without ever referring to coordinates, you avoid all these problems. The structure of problems just becomes a lot easier that way.
Other than that, a 4D Minovski space obviously is good for special and general relativistic concepts and you can also go for a conformal description of that space which gives you a 6D working space of which only 4 Dimensions have direct physical relevance.
It might be surprising, but adding in those two extra dimensions actually makes things easier, combining notions that shouldn't be separate at all, separating notions that shouldn't be confused and making transformations easier that are quite relevant in the physical world.
If you, however, are talking about higher dimensional spaces in which each of the dimensions is supposed to have physical relevance, then there might not be much to that in practice.
Unless, that is, you count statistical data as physically relevant: Such data can have a virtually infinite number of dimensions. And often it's necessary to transform data sets to be "shaped better", such that algorithms which, for instance, search for optima or structures in your data behave better.
Beyond that, even for naive 3D Space, without a homogeneous, projective or conformal extension, thinking of rotations to act in planes has certain advantages.
Note, however, that the conformal model is actually closer to the physical reality than the naive form. - At the very least you need a homogeneous model.
The projective part that deals with a point at infinity you might argue against by saying that in reality there is no such thing as a point that is infinitely distant. (I personally wouldn't argue that) However, even then, that addition is one of convenience. It just makes things so much simpler to treat translation and rotation; circles and lines; planes and spheres as one.
And by combining the two you end up with a system you can manipulate without ever referring to coordinates, which definitely is very natural and much closer to reality.