  Weakly interacting massive particles (WIMPs) with electroweak charges, such
as the wino and the Higgsino, stand out as natural candidates for dark matter
in the universe. In this paper, we study the search for WIMPs at future
multi-TeV $\mu^+\mu^+$ colliders. We investigate both the direct production
search of WIMPs through the mono-muon channel and the indirect search through
quantum corrections in elastic $\mu^+\mu^+$ Moller scattering. We find that the
indirect search has an advantage over the direct search with sufficient
luminosities, $\mathcal{O}(\text{ab}^{-1})$, and low systematic uncertainties,
$\lesssim 0.3\,\%$. This advantage arises due to the weaker mass dependence
observed in the indirect search in comparison to direct production methods. The
advantage is further enhanced if the initial muon beams are polarized.
Specifically, we demonstrate that the indirect search method can detect the
thermal mass target for the wino and the Higgsino for $\sqrt{s} =
6\,\text{TeV}$ and $2\,\text{TeV}$ (with $\sqrt{s}$ being the center of mass
energy), respectively, with $10\,\text{ab}^{-1}$, an $80\,\%$ polarized beam
and an accuracy of $0.1\,\%$. Our findings illuminate the potential of future
high-energy $\mu^+\mu^+$ colliders in advancing our understanding of dark
matter.
