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