The question is in the title exactly as I want to ask it, but let me provide some background and motivation.

Many of the properties of fields studied in the algebraic theory of quadratic forms are manifestly *elementary* properties in the sense of model theory: that is, if one field has this property, then any other field which has the same first-order theory in the language of fields has that same property. Examples:

being quadratically closed, being formally real, being real-closed, being Pythagorean (sum of two squares is always a square), for any fixed positive integer n, having I^n = 0 (follows from the Milnor conjectures!), the u-invariant, the level, the Pythagoras number...

These properties imply that at least for some fields $K$, if $L$ is any field elementarily equivalent ot $K$, then $W(L) \cong W(K)$: e.g. $K$ is quadratically closed, $K$ is real-closed, $K = \mathbb{C}((t))$. Is it always the case that $K \equiv L$ implies $W(K) \cong W(L)$? I am pretty sure the answer is no because for instance if $\operatorname{dim}_{\mathbb{F}_2} K^{\times}/K^{\times 2}$ is infinite, I think it is not an elementary invariant. And if you take a field with vanishing Brauer group, then $W(K)$ is, additively, an elementary $2$-group of dimension $\operatorname{dim}_{\mathbb{F}_2} K^{\times}/K^{\times 2} + 1$.

But are there known positive results in this direction?