Do I understand correctly that a "complete orthonormal set" is the same as basis?
Complete meaning that every Cauchy sequence in the set converges to an element in the set.
But then if you consider $l^2$, can't I choose the set containing $e_1,e_3,e_4,...$ e.g. skipping over the second standard basis element. This is orthonormal and seems complete, but no infinite linear sum of the elements converges to $e_2$, and then it is not a basis. Is there something wrong with the reasoning?
I think you are confusing two meanings of "complete". An orthonormal set (in an infinite dimensional normed vector space) is complete just when every vector is a(n infinite) sum of scalar multiples of vectors from the set.
Strictly speaking, those complete orthonormal sets are not bases, since a set can be a basis only if every vector is a finite sum of multiples of set elements.
In any case the "complete" in "complete orthonormal set" is not the same as a complete set in the topological sense. There what you are calling complete is that the closure of the set is the whole space.