(Question:) Suppose that $H$ is a Hilbert space over the field $\mathbb{C}$ and that $(v_n)_{n=1}^\infty$ is a Hilbert basis for it, that is a sequence of orthonormal vectors whose span is dense in $H$. I am wondering whether we can turn the sequence of real and imaginary parts of every $v_n$ also to a Hilbert basis for $H$, that is can we turn $(\mathrm{Re}(v_n), \mathrm{Im}(v_n))_{n=1}^\infty$ to an orthonormal sequence whose span is also dense in $H$? If you are wondering that the real and imaginary parts for $v_n$s mean, suppose for example that $H$ is some $L^2$ (or $l^2$) space over the field $\mathbb{C}$ so that each $v_n$ is a priori a complex valued function satisfying $v_n(x) = \mathrm{Re}(v_n(x)) + i\mathrm{Im}(v_n(x)) = r_n(x) + ic_n(x)$ for some real-valued functions $r_n, c_n$.
(Motivation:) We know from basic Fourier theory that the functions $$e_n(x) = \exp(2\pi i n x/(b-a))$$ form an orthonormal basis for the space $L^2([a,b];\mathbb{C})$. Take e.g. for $a = -\pi, b = \pi$. As $L^1([a,b];\mathbb{C})\subset L^2([a,b];\mathbb{C})$, we can represent any $f\in L^1([-\pi,\pi];\mathbb{C})$ as $$f(x) = \sum_{n=-\infty}^\infty \alpha_ne_n(x), \alpha_n\in\mathbb{C}$$
where $\alpha_n = \hat{f}(k)$. By writing $$\alpha_n = a_n + b_ni$$ with $a_n,b_n\in\mathbb{R}$, we can further write
$$\alpha_ne_n(x) = (a_n + b_ni)\cos(xn) + (a_ni - b_n)\sin(xn)$$
showing that $f$ is the sum over some real valued functions with complex coefficients. In the case of such $L^2$ space over a symmetric interval, sine and cosine have good orthogonality properties, that with $L^2$ normalization constants $s_n, t_n$, the sequence $(s_n\sin(xn), t_n\cos(xn))_{n=1}^\infty$ is an orthonormal sequence spanning $L^2([-\pi,\pi])$.
For a general such $L^2$ish space where we can meaningfully talk about the real and imaginary parts of the basic vectors, say, $(v_n)_{n=1}^\infty$, I think that it might not be true that immediately that elements of the sequence $(\mathrm{Re}(v_n), \mathrm{Im}(v_n))_{n=1}^\infty$ are orthogonal to one another. Note: I am saying I think, because I don't know how to construct a counter example. But in any case, provided that every elements of our general Hilbert space $f\in H$ can be written as
$$f = \sum_{n=1}^\infty \alpha_nv_n, \alpha_n\in\mathbb{C}$$
we may again decompose the sum to see that
$$f = \sum_{n=1}^\infty (a_n + b_ni)\mathrm{Re}(v_n) + (a_ni - b_n)\mathrm{Im}(v_n)$$
where $\mathrm{Re}(v_n)$ and $\mathrm{Im}(v_n)$ are real-valued. One idea to turn any finite set $\{w_1,\dots,w_n\}$ of possibly linearly dependent vectors $w_k$ to a new orthonormal set $\{u_1,\dots,u_j\}$, where $j\leq n$, such that the span of $\{w_1,\dots,w_n\}$ equals the span of $\{u_1,\dots,u_j\}$, is to use the Gram-Schmidt algorithm. Specifically, we are using Gram-Schmidt in such a way that we discard any zero elements that might occur when we subtract projections (to avoid division by zero), and instead move to the next element in our list of unprocessed vectors $w_l$.
(Problem(s):) Relabel the sequence $(\mathrm{Re}(v_n), \mathrm{Im}(v_n))_{n=1}^\infty$ to $(w_n)_{n=1}^\infty$. When trying to orthonormalize the sequence $(w_n)_{n=1}^\infty$, I ran into the difficulty of what to do if some $w_k$ depends linearly on arbitrarily many $w_l$s with $l > k$. To be more precise, given a new vector $w_k$ that is linearly dependent on other members of $(w_n)_{n=1}^\infty$, I am not sure how I can argue that
$$w_k = \sum_{j\in \mathcal{I}}\alpha_jw_j$$
for a finite index set $\mathcal{I}$. But even if such finiteness property would hold, I am not really sure what we should do if the $w_k$ elements would satisfy e.g. the property that for any $k\in\mathbb{N}$ there is a non-empty index set $\mathcal{I}(k)$ such that $w_k$ is in the linear span of the elements $w_l$ with $l\in\mathcal{I}(k)$ and $l >k$. In the finite dimensional case we don't have this sort of a problem, since there is an element in an ordered version of the set $\{w_1,\dots,w_n\}$. Is there any way to use G-S in this kind of an infinite dimensional Hilbert space?