Cosmetic Appliance Base

FIG. 1 is a front, right, top perspective view of a cosmetic appliance base showing the new design. FIG. 2 is a front elevation view thereof. FIG. 3 is a rear elevation view thereof. FIG. 4 is a left side elevation view thereof. FIG. 5 is a right side elevation view thereof. FIG. 6 is a top plan view thereof; and, FIG. 7 is a bottom plan view thereof. The broken lines in the drawings depict portions of the cosmetic appliance base that form no part of the claimed design.