The Dirac operator on an $N$-dimensional Euclidean-signature Riemann manifold $(M,g)$ is a bit more compicated than you make out. It takes the form
$$
{\mathcal D}= \gamma^a e^\mu_a \left(\partial_\mu + \textstyle{\frac 12} \sigma^{bc}\,\omega_{bc\mu}\right)= \gamma^a D_a.
$$
Here the
${\bf e}_a\equiv e_a^\mu\partial_\mu$ compose an orthonormal vielbein on $M$, the $\gamma^a$ are Hermitian matrices obeying
$$
\{\gamma^a,\gamma^b\}= 2\delta^{ab},
$$
the
$$
\sigma^{ab} = \textstyle{\frac 14} [\gamma^a,\gamma^b]
$$
are the skew-Hermitian spinor generators of $\mathfrak {so}(N)$ which obey
$$
[\sigma^{ab},\sigma^{cd}]= \delta^{bc} \sigma^{ad}- \delta^{ac} \sigma^{bd}+\delta^{ad} \sigma^{bc}-\delta^{bd} \sigma^{ac},
$$
and
$$
D_a \equiv D_{{\bf e}_a}=e^\mu_a \left(\partial_\mu + \textstyle{\frac 12} \sigma_{bc}\,{ \omega^{bc}}_\mu\right)
$$
is the covariant derivative acting on the components of a Dirac spinor. The ${ \omega^{bc}}_\mu$ are sometimes called the "spin connection" although they are just the components Levi-Civita connection in the vielbein frame.
The square of the Dirac operator is given by Lichnerowicz' identity
$$
({\mathcal D})^2= \nabla^2 - \frac 14 R,
$$
where
$$
\nabla^2\equiv\frac 1{\sqrt{g}} D_\mu \sqrt{g}g^{\mu\nu} D_\nu
$$
is the "rough'', or "connection'' Laplacian acting on spinors so
$$
D_\mu= \left(\partial_\mu + \textstyle{\frac 12} \sigma_{bc}\,{ \omega^{bc}}_\mu\right).
$$
There are (not short) proofs of Lichnerowicz' result elsewhere on Stack Exchange, but I have no time to find them for you.
I think for other Laplace equation you may like is the Hodge laplacian
$$
(\delta+d)^2=-\nabla^2_{\rm Hodge}
$$
which involves the exterior derivative $d$ and its adjoint $\delta$ which act on differential forms. It is not a simple Laplacian but is related to it via the Bochner-Weitzenboeck identity. Like the Lichnerowicz LaplacIan, the Hodge Laplacian has a square root
$$
D=d+\delta
$$ whose action can be made to look like Dirac's.
Write
$$
\psi_{\alpha\beta}= \sum_{p=0}^d \frac 1 {p!} A_{\mu_1\mu_2\cdots \mu_p}(x)(\gamma^{\mu_1} \ldots \gamma^{mu_p})_{\alpha\beta}
$$
If we let $d+\delta$ act on the Euclidean space forms
$$
\frac 1 {p!} A_{\mu_1\mu_2\cdots \mu_p}(x)dx^{\mu_1}\ldots dx^{\mu_p}
$$
in the usual way, then the action of the two-spinor $\psi_{\alpha\beta}$ is
$$
\psi_{\alpha\beta}\mapsto [(d+\delta)\psi]_{\alpha\beta}=(\gamma^\mu)_{\alpha\beta'} \partial_\mu \psi_{\beta'\beta}.
$$
Here, in flat space only, $\beta$ acts rather as a spectator. These fermions are called Kahler-Dirac fermions, or in a natural lattice version, Kogut-Susskind staggered fermions. In the lattice version the $\beta$ indices are called "tastes" are are regarded as "fermion-doubling" lattice artifacts. I believe that Kahler-Dirac fermions were introduced by Edington --- although I have no reference for this. I do not know of anywhere that they exist in nature, but they have a large footprint in the theory literature.
