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Rechargeable solid-state magnesium batteries are considered for high energy density storage and usage in mobile applications as well as to store energy from intermittent energy sources, triggering intense research for suitable electrode and electrolyte materials. Recently, magnesium borohydride, Mg(BH$_{4}$)$_{2}$, was found to be an effective precursor for solid-state Mg-ion conductors. During the mechanochemical synthesis of these Mg-ion conductors, amorphous Mg(BH$_{4}$)$_{2}$ is typically formed and it was postulated that this amorphous phase promotes the conductivity. Here, electrochemical impedance spectroscopy of as-received γ-Mg(BH$_{4}$)$_{2}$ and ball milled, amorphous Mg(BH$_{4}$)$_{2}$ confirmed that the conductivity of the latter is ~2 orders of magnitude higher than in as-received γ-Mg(BH$_{4}$)$_{2}$ at 353 K. Pair distribution function (PDF) analysis of the local structure shows striking similarities up to a length scale of 5.1 Å, suggesting similar conduction pathways in both the crystalline and amorphous sample. Up to 12.27 Å the PDF indicates that a 3D net of interpenetrating channels might still be present in the amorphous phase although less ordered compared to the as-received γ-phase. However, quasi elastic neutron scattering experiments (QENS) were used to study the rotational mobility of the [BH$_{4}$] units, revealing a much larger fraction of activated [BH$_{4}$] rotations in amorphous Mg(BH$_{4}$)$_{2}$. These findings suggest that the conduction process in amorphous Mg(BH$_{4}$)$_{2}$ is supported by stronger rotational mobility, which is proposed to be the so-called “paddle-wheel” mechanism.