When I say $\mathbb{C}$ is a number system over $\mathbb{R}^{2}$, I mean that the field of complex numbers is identified with the set of dilative rotations of the affine real plane ($\mathbb{R}^{2}$) characterized by the Pythagorean metric. When I say $\mathbb{H}$ is a number system over $\mathbb{R}^{4}$, I mean that the skew-field of quaterions is identified with the set of dilative Hermitian rotations over the affine complex plane $\mathbb{C}^{2}$ characterized by the Hermitian metric. When I say these are continuous, I mean that they inherit the continuum properties of the real numbers.
It occurs to me that Minkowski space is a geometry over $\mathbb{R}^{4}$ characterized by the Minkowski metric. But the continuum of spacetime events does not appear to constitute a number system. Unlike the complex numbers and the quaternions, the Lorentz transformations and space points (spacetime events) are "decoupled".
So I ask, is there some theorem or uncontradicted proposition that says the only continuous number systems expressible as ordered pairs and ordered quadruples of real numbers are the complex numbers and quaternions, respectively?