In this work, we present an approach for monocular hand-eye calibration from per-sensor ego-motion based on dual quaternions. Due to non-metrically scaled translations of monocular odometry, a scaling factor has to be estimated in addition to the rotation and translation calibration. For this, we derive a quadratically constrained quadratic program that allows a combined estimation of all extrinsic calibration parameters. Using dual quaternions leads to low run-times due to their compact representation. Our problem formulation further allows to estimate multiple scalings simultaneously for different sequences of the same sensor setup. Based on our problem formulation, we derive both, a fast local and a globally optimal solving approach. Finally, our algorithms are evaluated and compared to state-of-the-art approaches on simulated and real-world data, e.g., the EuRoC MAV dataset.