In the 1860s, Maxwell published his work describing electricity and magnetism within a single framework. Lorentz, Heaviside, and others refined his results, and voilà: Electromagnetism.
Ørsted's observation can be seen as one starting point for the trend towards "unification": the idea that multiple phenomenon that at first seem unrelated may in fact be aspects of a single, underlying subject.
This is one of the major themes of modern physics. All from a little twitch of a compass needle.