Background
Our question concerns the generalisation of Bloch sphere rotations to higher-dimensional Bloch spheres. We note the connection between states of a Hilbert space represented on $S^2$ with the global phase invariance corresponding to the fibre space of the complex Hopf fibration and wish to find some representation of states on the base space of the quaternionic Hopf fibration of $S^4$ which we consider as the next order of Bloch sphere, also noted in Mosseri and Dandoloff, however now in terms of states in $\mathbb{C}^4.$
We begin with describing general rotation of the $S^2$ Bloch sphere generated by the Pauli-matrices and wish to find the higher-dimensional generalisation of this to the $S^4$ Bloch-sphere.
The Pauli-matrices can be used to generate an arbitrary rotation of the Bloch-sphere "around the axis defined by the corresponding $\sigma^k$", written in terms their eigenvectors. These are typically written as
$$Z(\alpha) := |0\rangle\langle 0| + e^{i \alpha}|1\rangle\langle 1| \qquad\qquad X(\alpha) := |+\rangle\langle +| + e^{i \alpha}|-\rangle\langle -|$$ with $Z(\alpha)$ corresponding to a rotation about the axis defined by $\sigma^3$ (the "z-axis") and $X(\alpha)$ by $\sigma^1$ (the "x-axis"). Composing these, we can write any arbitrary rotation of the Bloch sphere.
Question
Given that we can interpret the Pauli-matrices as the three axes of $S^2$ and it is possible to construct arbitrary rotation around these using a single angle relative to these axes, is there a general notion of how to extend this notion to the four axis of $S^4$?
As an aside, being that the quaternionic Hopf fibration is usually written in terms of vectors $v \in \mathbb{H}^2$ the global phase freedom, encoded by the fibre space of $S^3$ in $S^7$, in this case would constitute a phase freedom of one quaternion. In the case of $\mathbb{C}^4$, how can this global phase invariance be interpreted?
Guiding attempt
We wish to define four rotation operators corresponding to each of the four axes of the space of $S^4$, rotations around which should be parameterised by a single phase argument.
A basis for $\mathbb{C}^4$ can be generated by taking tensor products between the eigenvectors of $\sigma^3$. We label these new basis vectors by $|00\rangle, |01\rangle, |10\rangle, |11\rangle$. Placing relative angles on these, we get the following rotation operator parameterised by three phases.
$$ T(\alpha,\beta,\gamma) := |00\rangle\langle 00| + e^{i \alpha}|01\rangle\langle 01| + e^{i \beta}|10\rangle\langle 10| + e^{i \gamma}|11\rangle\langle 11| $$
But this cannot be a good generalisation of because we have three angles for one axis. We also note that we have two sets of two basis vectors with degenerate eigenvalues ($\pm 1$) so will instead pair the vectors with degenerate eigenvalues into two sets and give a relative phase between these two sets as below.
$$T(\alpha) := |00\rangle\langle 00| + |01\rangle\langle 01| + e^{i \alpha}(10\rangle\langle 10| + |11\rangle\langle 11|)$$
These might be recognised as the eigenvectors of $\gamma^0$ of the $\gamma$-matrices. Is this a sound construction with regard to the notions mentioned above? Can this construction in be interpreted as rotation around an axis in $S^4$? Is it possible to extend this construction to the remaining $\gamma$-matrices to generate arbitrary rotations around axes in $S^4$?
Thank you in advance for reading and hopefully being able to give us some insight into this general construction.