Research on digital communication receivers based on 1-bit quantization and oversampling has gained momentum as high resolution in time domain is less difficult to achieve than high resolution in amplitude domain. However, as 1-bit quantization is a highly non-linear operation, standard receiver algorithms cannot be applied.In this work we consider maximum likelihood (ML) carrier phase estimation under a white noise assumption. We show that in the low signal-to-noise ratio (SNR) regime least squares (LS) phase estimation is equivalent to ML phase estimation. Subsequently, we derive the expectation-maximization and the scoring algorithm to solve the ML problem iteratively for any SNR, where the LS estimate can be used as initialization. We evaluate the performance numerically and see that both algorithms converge to the ML solution but the scoring algorithm converges much faster with two steps being sufficient. We further observe that colored noise improves the performance compared to white noise.