No more than any other human - at least at first. Her apparent ability to die & then reappear alive in another spacetime location was explained in "The Name of the Doctor", which revealed that there have been many (according to her voiceover, thousands) of "echoes" of her throughout the Doctor's timeline. She also gained super-hacking abilities in "The Bells of Saint John" that, while technically temporary, were still retained in some respect as Clara was able to adapt very quickly to new technology even after only being shown how to use it once (or even not at all) such as alien vehicles ("The Rings of Akhaten"), Capt. Jack Harkness's vortex manipulator ("The Day of the Doctor"), not to mention by the time "Hell Bent" rolls around she has learned how to fly a TARDIS (whilst her immortal friend, Ashildr, is still learning from the manual and can't even figure out the chameleon circuit). In addition, after her extraction in "Hell Bent" she is functionally immortal, does not age, and is likely indestructible as the universe cannot let her die until she returns to the trap street; as a result, as of that episode, the answer becomes yes, she does have some degree of superhuman abilities based upon her time-locked status.