We propose a new quantile regression model when data are subject to censoring. In comparison with some existing approaches, our model requires neither any global linearity assumption, nor unconditional independence of the survival and the censoring time. We develop a class of power-transformed quantile regression models such that the transformed survival time can be better characterized by linear regression quantiles. Consistency and asymptotic normality of the resulting estimators are shown. A re-sampling based approach is proposed for statistical inference. The finite sample performance of the new estimator is studied by extensive simulation studies and a real data analysis.