The aim of this article is to present a novel quaternion based control scheme for the attitude control problem of a quadrotor and experimentally evaluate its performance. A quaternion is a hyper complex number of rank 4 that can be utilized to avoid the inherent geometrical singularity when representing rigid body dynamics with Euler angles or the complexity of having coupled differential equations with the Direction Cosine Matrix (DCM). In the presented approach the novel contributions consist of: a) the quadrotor's attitude model and b) the proposed non-linear Proportional squared (P²) control algorithm, which have been proposed and experimentally evaluated fully in the quaternion space, without any transformations nor calculations in the Euler angle nor the DCM spaces. The established control scheme is combined with a quaternion based Madwick Complementary filter for estimating the attitude quadrotor's responses. Multiple experimental results, including the case where external disturbances are acting on the quadrotor, are being presented for proving the efficiency and the robustness of the proposed novel quaternion based controller.