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When modelling survival data, it is common to assume that the survival time T is conditionally independent of the censoring time C given a set of covariates. However, there are numerous situations in which this assumption is not realistic. The goal of this paper is therefore to develop a flexible semiparametric transformation model, which states that after a proper nonparametric monotone transformation, the vector (T,C) follows a linear model, and the vector of errors in this bivariate linear model follows a bivariate normal distribution with non-diagonal variance-covariance matrix. We show that this semiparametric model is identified, and propose estimators of the nonparametric transformation, the regression coefficients and the correlation between the error terms. The performance of the proposed method is investigated both in an asymptotic way and through finite sample simulations.