If a basis set is complete, are the elements in it mutually orthonormal? For example, we can express the field operator in the basis of the creation and annihilation operators.This basis is complete, are the elements in it mutually orthonormal?
The answer is NO
Given any complete basis, one can construct another by taking independent combinations.
Assuming the original basis is complete and orthogonal, and contains $V_1$ and $V_2$, (which are thus orthogonal).
Replacing $V_2$ by $V_2'=V_1+V_2$ does not change the completude of the basis. But now two vectors of the new basis, namelt $V_1$ and $V'_2$ are not orthogonal.
If the questio is, whether it is always possible to find an orthogonal basis is one thing. But there is no need for a basis to be orthogonal to be complete.