We present the results of Chandra X-ray Observatory observations of the transient anomalous X-ray pulsar (AXP) candidate AX J1845.0-0258 in apparent quiescence. Within the source's error circle, we find a point source and possible counterpart, which we designate CXOU J184454.6-025653. No coherent pulsations are detected, and no extended emission is seen. The source's spectrum is equally well described by a blackbody model of temperature kT~2.0 keV or a power law model with photon index Gamma~1.0. This is considerably harder than was seen for AX J1845.0-0258 during its period of brightening in 1993 (kT~0.6 keV) despite being at least ~13 times fainter. This behavior is opposite to that observed in the case of another transient AXP, XTE J1810-197. We therefore explore the possibility that CXOU J184454.6-025653 is an unrelated source, and that AX J1845.0-0258 remains undetected since 1993, with flux 260-430 times fainter than at that epoch. If so, this would represent an unprecedented range of variability in AXPs.