This paper presents a novel morphing quadrotor, named MorphoCopter, covering its design, modeling, control, and experimental tests. It features a unique single rotary joint that enables rapid transformation into an ultra-narrow profile. Although quadrotors have seen widespread adoption in applications such as cinematography, agriculture, and disaster management with increasingly sophisticated control systems, their hardware configurations have remained largely unchanged, limiting their capabilities in certain environments. Our design addresses this by enabling the hardware configuration to change on the fly when required. In standard flight mode, the MorphoCopter adopts an X configuration, functioning as a traditional quadcopter, but can quickly fold into a stacked bicopters arrangement or any configuration in between. Existing morphing designs often sacrifice controllability in compact configurations or rely on complex multi-joint systems. Moreover, our design achieves a greater width reduction than any existing solution. We develop a new inertia and control-action aware adaptive control system that maintains robust performance across all rotary-joint configurations. The prototype can reduce its width from 447 mm to 138 mm (nearly 70\% reduction) in just a few seconds. We validated the MorphoCopter through rigorous simulations and a comprehensive series of flight experiments, including robustness tests, trajectory tracking, and narrow-gap passing tests.