For the usual $4$-dimensional Minkowski space $M$, the standard Dirac operator is given by $$ D: C^{\infty}(M) \to C^{\infty}(M), ~~~~~ f \mapsto \sum_{i=1}^4 \gamma_i\frac{\partial f}{\partial x_i}, $$ where the $\gamma_i$ are the usual gamma matrises. As we all know, $D$ was originally constructed as a square root of the Lapalcian $\sum_{i=1}^4 \frac{\partial^2 }{\partial x_i^2}$. Indeed, routine calculation will verify that $D$ does indeed square to give the Laplacian.

Moreover, there exists another square root of the Laplacian, namely $$ d + \ast d \ast: C^{\infty}(M) \to \Omega^1(M), $$ where $d$ is the usual exterior derivative, and $\ast$ is the Hodge $\ast$-operator.

How do these two square roots of the Laplacian relate to each other? Are they two separate objects, or just two ways of looking at the same thing?