  We present here several versions of the Grothendieck inequality over the skew
field of quaternions: The first one is the standard Grothendieck inequality for
rectangular matrices, and two additional inequalities for self-adjoint
matrices, as introduced by the first and the last authors in a recent paper. We
give several results on ``conic Grothendieck inequality'': as Nesterov
$\pi/2$-Theorem, which corresponds to the cones of positive semidefinite
matrices; the Goemans--Williamson inequality, which corresponds to the cones of
weighted Laplacians; the diagonally dominant matrices. The most challenging
technical part of this paper is the proof of the analog of Haagerup result that
the inverse of the hypergeometric function $x {}_2F_1(\frac{1}{2}, \frac{1}{2};
3; x^2)$ has first positive Taylor coefficient and all other Taylor
coefficients are nonpositive.
