We prove sufficient conditions for the boundedness of the maximal operator on variable Lebesgue spaces with weights φt,γ(τ)=|(τ−t)γ|, where γ is a complex number, over arbitrary Carleson curves. If the curve has different spirality indices at the point $t$ and γ is not real, then φt,γ is an oscillating weight lying beyond the class of radial oscillating weights considered recently by V. Kokilashvili, N. Samko, and S. Samko.