At the end of the 19th century, Hamilton’s quaternions already constituted a valid alternative to the Gibbs and Heaviside formalism to represent the algebra of three-dimensional space. But a black cloud loomed on the horizon: the fact that the square of the bases of the quaternions was negative. Maxwell noted that, since kinetic energy is a quadratic form, expressing it with quaternions, would have resulted a negative quantity: unacceptable!
Furthermore, a further step occurred in 1908 when Minkowski, to express the calculations of the newborn theory of relativity, preferred to extend the Gibbs trivectors rather than use the quaternion formalism, which he considered too restrictive in the expression of the Lorentz boost.
It is interesting to note that this difficulty lies in the fact that the square of the bases of the quaternions is negative: the same difficulty highlighted by Maxwell years earlier.
A further nail on the coffin of the quaterions was placed in 1927, when the formalism of the newborn quantum mechanics was born, which formalized the evolution of the wave function with a complex spinor. And that’s a real shame, considering that spinors are very close relatives of quaternions!
In short, it was true that quaternions were just a three-dimensional extension of complex numbers that simplified rotations in space, but they weren’t enough to fully describe physical space.