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A hyperk\"ahler manifold $M$ has a family of induced complex structuresindexed by a two-dimensional sphere $S^2 \cong \mathbb{CP}^1$. The twistorspace of $M$ is a complex manifold $Tw(M)$ together with a natural holomorphicprojection $Tw(M) \to \mathbb{CP}^1$, whose fiber over each point of$\mathbb{CP}^1$ is a copy of $M$ with the corresponding induced complexstructure. We remove one point from this sphere (corresponding to one fiber inthe twistor space),and for the case of $M = \mathbb{R}^{4n}$, $n\in\mathbb{N}$,equipped with the standard hyperk\"ahler structure, we construct onequantization that replaces the family of Berezin-Toeplitz quantizationsparametrized by $S^2-\{ pt\}$. We provide semiclassical asymptotics for thisquantization.

The twistor space of ${\mathbb{R}}^{4n}$ and Berezin-Toeplitz operators