I want to know if there exists an (real or complex) vector space $X$ with infinite dimension and an inner product $\langle\cdot,\cdot\rangle$ such that there is no orthogonal Hamel (algebraic) basis of $X$. I do not seek conditions on to the topological aspects of completeness nor an example Schauder Basis or a Maximal Orthogonal System in Hilbert Spaces. I am aware that this is not usually discussed in the scenario of infinite dimensional vector spaces, but this is exactly what I'm curious about.
I know how to prove that every inner product space of Hamel dimension $\aleph_0$ has an orthonormal Hamel basis using the Gram-Schimidt process but this proof does not work for the uncountable case.
I also know that a maximal orthogonal set of non-zero vectors is not always a Hamel basis, thus a simple application of Zorn's Lemma does not seem to solve the problem. Finally, I have seen many other similar questions here but none of them provide an answer for the question posed here.