The Sorting Hat is significantly different from most other enchanted objects we've seen in the Wizarding World. The hat is sapient - it not only converses with Harry on multiple occasions but invents new songs each year, one of which specifically states that "The founders put some brains in [it]". It obviously has complex reasoning and problem-solving skills in order to carry out its main purpose of sorting the first-years into their appropriate Houses. The songs it writes and performs deal with current events, and it even communicates in modern English, when it would have originally spoken in the very different dialect of ~1000 years ago, so it can learn and adapt.
We know that many enchantments cease the moment the caster dies, and not just ones that require continued concentration to use (like Imperio) but those "set it and forget it" types (like Petrificus Totalus, or, if you consider the movies to be A-Canon, the goldfish).
In contrast to other enchanted objects that continue to operate after the death of the witch or wizard who created them, wizarding portraits obviously have some semblance of sentience, but it is known to be limited. Wizarding photographs are not interactive at all, and only move more like a looping video.
Horcruxes exhibit a large amount of sentience, most notably Tom Riddle's diary. It is able to communicate with both Ginny and Harry, able to pull Harry into a Pensieve-like experience, and eventually even generate a close-to-corporeal form. It, too, is able to learn, adapt, and even feel, given that it finds out about Harry and desired to know more about him. Also, by their very nature, Horcruxes live on after the caster should have died.
Horcruxes are considered very dark magic in the modern era, requiring rending the soul via murder and evil incantations to contain a part of it inside an object. However, 1000 years ago, when the founders of Hogwarts lived, this could have been considered less evil, even normal, perhaps. Godric Gryffindor owned a sword, an object only used for battle. It is not a stretch of the imagination that he could have killed with that sword, (or with his wand).
Is the Sorting Hat a Horcrux?