$\operatorname{spin}^c$ structure through the Pauli matrices

Question

Let $M$ be a n-dimensional compact oriented smooth manifold. As far as I know, a $\operatorname{spin}^c$ structure on $M$ (or $TM$) is either a principal $\operatorname{spin}^c(n)$-bundle $P$ such that the associated vector bundle through the adjoint representation $\operatorname{spin}^c(n)\to SO(n)$ is isomorphic to $TM$ or a representation $\gamma:TM\to End(W)$ such that $$\gamma(v)+\gamma(v)^*=0 \qquad \& \qquad \gamma(v)\gamma(v)^*=|v|^2Id$$ where $W$ is an hermitian bundle of rank $m=2^{n/2} \quad or \quad 2^{(n-1)/2}$ depending on which is an integer. It came to my attention by reading https://arxiv.org/abs/dg-ga/9601007 that in the 3-dimensional case, the second definition can be rephrased in terms of the Pauli matrices. Precisely, there exists an orthonormal basis $\{e_1,e_2,e_3\}$ of $TM$ and a similar one on $W$ such that $$\gamma(e_1)=\begin{pmatrix}0&1\\1&0\end{pmatrix}\qquad \gamma(e_2)=\begin{pmatrix}0&-i\\i&0\end{pmatrix}\qquad \gamma(e_3)=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ However, as hermitian matrices they do not satisfy the first condition because the involution of each hermitian matrix is the matrix itself. Am I missing something? The reason of this should come from the representation of the purely imaginary quaternion over $\mathbb C^2$ with which one constructs the Clifford Algebra and, in turn, the $\operatorname{spin}^c$ group but after playing around with the matrices I couldn't obtain the Pauli matrices.