The Stern Gerlach experiment in 3-Dimensions provides us with conditions on what properties Spin-Half vectors must satisfy, from which we can build our basis states in $x, y, \text{ and } z$. The conditions can be expressed as follows (where $i$ and $j$ are variables representing spatial dimensions):
$$\ \langle \pm_i | \pm_i \rangle=1, \langle \pm_i | \mp_i \rangle=0$$
$$\text{For } i \neq j : \big|\langle \pm_i | \pm_j \rangle \big|^2= \big| \langle \pm_i | \mp_j \rangle \big|^2=\frac{1}{2}$$
Of course there are infinitely many solutions to the 6 basis vectors, but as a convention we define:
$$|+_z\rangle = \begin{pmatrix} 1\\ 0 \\ \end{pmatrix}, |-_z\rangle = \begin{pmatrix} 0\\ 1 \\ \end{pmatrix}, |\pm_x\rangle = \frac{|+_z\rangle \pm |-_z\rangle }{\sqrt2}, |\pm_y\rangle = \frac{|+_z\rangle \pm i|-_z\rangle }{\sqrt2}$$
All other solutions can be seen as rotations of this convention about the Bloch Sphere.
From here it is clear that were this experiment only being done in 2-Dimensions, then complex numbers would not be required to define the vectors, since the $z$ and $x$ basis span 2-Dimensions and they can be constructed with only real numbers. Which leads me to wonder; what algebra would we have to use if we were in 4 spatial dimensions performing the Stern-Gerlach Experiment. Our conditions would be expressed identically as:
$$\ \langle \pm_i | \pm_i \rangle=1, \langle \pm_i | \mp_i \rangle=0$$
$$\text{For } i \neq j : \big|\langle \pm_i | \pm_j \rangle \big|^2= \big| \langle \pm_i | \mp_j \rangle \big|^2=\frac{1}{2}$$
However now $i$ and $j$ are indices over 4 spatial dimensions, let's say $w, x, y, z$. We need to define mathematical objects to represent our 8 basis vectors that satisfy these conditions; 6 of which is trivial, in that its identical to the 3-Dimensional case, but for the new dimension, can it be done with just complex numbers? Or do we have to resort to something else, potentially quaternions?
