I'm not quite sure about this but I wrote down during a lecture that a normed vector space with an algebraic basis containing a countably infinite number of elements is never complete.
What I mean by algebraic basis is that the elements of this vector space are the finite combinations of those from the basis.
I'm wondering whether the statement should actually contain "infinite" instead of "countably infinite" but I'm not even sure.
Can anyone tell me whether this statement is true or correct me if it's close to being true ?
Yes, it is true: if a normed vector space has a countable Hamel basis, then it cannot be complete. This is intuitively easy to grasp. Let $\{e_n\,|\,n\in\mathbb N\}$ be such a basis. You can assume, without loss of generality, that each $e_n$ has norm $1$. Consider the series $\sum_{n=1}^\infty\frac1{n^2}e_n$. If the space was complete, then this series would converge. But you cannot express its sum as a finite linear combination of the $e_n$'s.