I'm trying to understand the values that fields can take. For fermions, my understanding is that fields on spacetime take values as Dirac Spinors, which are $\mathbb{C}^4$ vectors. The vector space of Dirac Spinors is the one acted on by the matrix ring generated by the Gamma Matrices. Gamma Matrices are pairwise tensor products of Pauli Matrices, which generate the quaternions $\mathbb{H}$. Incidentally, the Clifford Algebra for spacetime $Cl_{1,3}(\mathbb{R}) \cong M(2,\mathbb{H})$, which is isomorphic to the matrix ring generated by the Gamma Matrices. $M(2,\mathbb{H})$ acts on $\mathbb{C}^4$, and therefore so do the Gamma Matrices. As a result, fields $\psi(x,t)$ for fermions on spacetime take values in the space of $\mathbb{C}^4$ spinors.
Taking a slightly more dubious route... The spacetime invariants for fermions under special relativity are given by the Minkowski Metric. The Clifford Algebra corresponding to the Minkowski Metric is $Cl_{1,3}(\mathbb{R}) \cong M(2,\mathbb{H})$, which acts on the vector space $\mathbb{C}^4$. Every relativistic field $\psi(x,t)$ on spacetime preserves the invariants of spacetime, and therefore $\psi(x,t)$ must take values in some subspace of $\mathbb{C}^4$ acted on by some sub-algebra of $Cl_{1,3}(\mathbb{R})$. For fermions, $\psi(x,t)$ takes values in the full subspace $\mathbb{C}^4$ itself.
I'm wondering, first, if that dubious route is valid. If it is valid, I'm wondering if, as a consequence, all relativistic fields on spacetime, even for bosons, take values in some subspace of $\mathbb{C}^4$ spinors.