It’s disingenuous to suggest that Hamilton was “wrong” to develop Quaternions and Clifford was “right”. Historically, Hamilton worked out his theory of quaternions apparently a year earlier (1843 as opposed to 1844), so that Clifford, in developing Geometric Algebra, had a hindsight unavailable when Hamilton described quaternions. Moreover, even Clifford did not suggest there is such a significant difference between quaternions and rotors, as is suggested in this video (see https://en.wikipedia.org/wiki/Geometric_algebra#History):

“From his (Clifford’s) point of view, the quaternions described certain transformations (which he called rotors), whereas Grassmann’s algebra described certain properties (or Strecken such as length, area, and volume).”

An issue that is pointed out in that wikipedia article is that for mechanics, the Grassmann/Clifford approach did not catch on, not because of quaternions per se, but because, as an extension of the Gibbs/Heaviside approach to vector analysis, quaternions seemed more natural to the applied mathematicians and physicists. The Grassmann/Clifford Approach is more of a pure mathematics approach to continuing the extensions to higher dimensional structures, in part because one must then give up not only commutativity but associativity of the attendant arithmetic as well. This requires a higher level of abstraction, and is typically avoided by those who tie their study of mathematics to applications.

By the way, rotors are just a representation for certain quaternions, so actually, quaternions are more general — they actually form a vector space, and not just a group (again see https://en.wikipedia.org/wiki/Geometric_algebra#History, as well as https://en.wikipedia.org/wiki/Rotor_(mathematics)):

“In mathematics, a rotor in the geometric algebra of a vector space V is the same thing as an element of the spin group Spin(V). We define this group below.”

versus

“The set of quaternions is made a 4-dimensional vector space over the real numbers, …”

Thus the quaternions don’t form a group because they form a toolbox that is larger than the one you are advocating to replace them, since the rotors form a group whose members are (up to isomorphism) the unit quaternions. The isomorphism maps the rotors (members of the Spin(3) group) one-to-one onto the (real) 3 dimensional (topological, actually, Lie) group of (2×2) special unitary matrices over the complex numbers. The (2×2) complex matrices form a 4 dimensional complex space, the quaternions can be represented as a two-dimensional complex subspace of this 4 dimensional complex vector space that is closed under matrix multiplication, for which the nonzero members all are invertible, and the rotors are represented by a subset of this complex vector space that is neither a complex vector space nor a real vector space, being not closed under either real or complex multiplication, and being not closed under matrix addition.

Now, if you claim that rotors are “better” because they are more efficient, I beg to differ. The notation for quaternions is not more complex than that for rotors. In fact, there is other internet discussion about this for game development (see https://news.ycombinator.com/item?id=18365433):

“Back in 2000 there was some debate about dropping quaternions in favor of a set of equivalent operations on plain matrices. https://www.gamedev.net/articles/programming/math-and-physic… The debate petered out when the proponents finished optimizing their implementation and found they had produced exactly the same code as the existing quaternion implementations. The only difference was the approach used to derive it.” (posted by corysama)


Aside:
Unfortunately, some posts to that site continue to illustrate the failure of many to understand the relationship between linear transformations and matrices, which is a mathematical error analogous to the well-known kinds of engineering errors that caused losses of mars rovers and satellites:

“Matrices are linear transformations.” (posted by certhas)

Why is this analogous to an engineering error? Well, given a linear transformation T on a real vector space, for example (even a one dimensional real vector space), there are infinitely many matrices that can represent that same linear transformation. One way to produce two different matrices representing T is to change the units used in describing the way one measures distance along the axes, because that results in a change of basis, and changes the entries of the matrix one uses to represent T. For example, one may use inches in one basis and millimeters in another, and this gives two different matrices representing T. The other way is problematic as well: Given a real matrix A, there are multiple linear transformations represented by A, even if we fix the real vector spaces upon which we allow A to act. In order to “allow A to act”, one must not only choose a vector space, but also a basis for that vector space. Once that is done, A represents a unique linear transformation, but if one changes the basis, even by changing the order of the vectors listed in that basis, the linear transformation represented by A is different for the new basis. This is the case, even for (1×1) matrices, and in this simple case, the entire issue is one of scaling, and hence the choice of units. Again, this view of linear transformation on a 1 dimensional space as “the same as a (1×1) matrix” is an engineering error, when the problem one is working on is an engineering problem with exactly one degree of freedom.

Why this aside? Well, my point all through here is that one needs to be precise with language and with what is meant by “better”, but before one can even come close to the proper level of precision with language about that concept, one must first accurately portray the mathematics and the history and give a careful description of what one means by “better”. This rhetorical mumbo jumbo that relies on misinterpretations of the facts, such as distinguishing rotors from quaternions when mathematically, rotors are up to isomorphism special types of quaternions, is inaccurate rhetoric in the same way that people say “oh well, a linear transformation is just a matrix”, when in fact, it jus’ ain’t so, sir.