I think that's a different point. To get rotations in 3D you don't need to go to 5D conformal GA.
To be able to unify many geometric objects, like lines and spheres and point pairs and represent the duality e.g. the "meet" of two lines constructs a point (possibly a point at infinity) and the "join" of two points constructs a line (not sure what happens if the two points are identical), .... then you need 5D which is really like 2^5 = 32D in my mind.
But if all you're trying to do is stop being confused by two different things that both look like vectors, but transform differently under spatial transformations (i.e. any vector that is the result of the cross-product is really a different type of vector than the argument vectors, or e.g. normal vectors) then 3D GA is fine. Though really it's more like 2^3 = 8D (1 scalar, 3 regular basis vectors, 3 "axial" basis vectors, and one pseudo scalar).
An efficient implementation would likely need to use a type system avoid representing 8 dimensions directly. Like the cross product of two vectors will produce an element where only the "axial" components are non-zero.
You are right, 5D conformal GA goes way beyond nice rotations in 3D space, but why stop there? We can have two worlds: computationally expensive, high level, powerful abstractions and computationally efficient, messy abstractions. In some cases of rotation, even quaternions are too much.
To be able to unify many geometric objects, like lines and spheres and point pairs and represent the duality e.g. the "meet" of two lines constructs a point (possibly a point at infinity) and the "join" of two points constructs a line (not sure what happens if the two points are identical), .... then you need 5D which is really like 2^5 = 32D in my mind.
But if all you're trying to do is stop being confused by two different things that both look like vectors, but transform differently under spatial transformations (i.e. any vector that is the result of the cross-product is really a different type of vector than the argument vectors, or e.g. normal vectors) then 3D GA is fine. Though really it's more like 2^3 = 8D (1 scalar, 3 regular basis vectors, 3 "axial" basis vectors, and one pseudo scalar).
An efficient implementation would likely need to use a type system avoid representing 8 dimensions directly. Like the cross product of two vectors will produce an element where only the "axial" components are non-zero.