Sorry bumping in here,

the chord remains a C chord without any additions as long as you stick with the notes of that chord, in this case it is from low to high C, E, G, C, G so no additional notes from the C scale are added.

Sometimes you might bump in tabs in which they refer to this chord as a C5 chord but that is technically not correct, a C5 chord is a so called power chord and technically not a real chord since power chords only have a root note and a fifth, no third which is clearly not the case here,

hope this helps you?

Herman