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We explore the transit timing variations (TTVs) of the young (22 Myr) nearby AU Mic planetary system. For AU Mic b, we introduce three Spitzer (4.5 $μ$m) transits, five TESS transits, 11 LCO transits, one PEST transit, one Brierfield transit, and two transit timing measurements from Rossiter-McLaughlin observations; for AU Mic c, we introduce three TESS transits. We present two independent TTV analyses. First, we use EXOFASTv2 to jointly model the Spitzer and ground-based transits and to obtain the midpoint transit times. We then construct an O--C diagram and model the TTVs with Exo-Striker. Second, we reproduce our results with an independent photodynamical analysis. We recover a TTV mass for AU Mic c of 10.8$^{+2.3}_{-2.2}$ M$_{E}$. We compare the TTV-derived constraints to a recent radial-velocity (RV) mass determination. We also observe excess TTVs that do not appear to be consistent with the dynamical interactions of b and c alone, and do not appear to be due to spots or flares. Thus, we present a hypothetical non-transiting "middle-d" candidate exoplanet that is consistent with the observed TTVs, the candidate RV signal, and would establish the AU Mic system as a compact resonant multi-planet chain in a 4:6:9 period commensurability. These results demonstrate that the AU Mic planetary system is dynamically interacting producing detectable TTVs, and the implied orbital dynamics may inform the formation mechanisms for this young system. We recommend future RV and TTV observations of AU Mic b and c to further constrain the masses and to confirm the existence of possible additional planet(s).