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We investigate the intersection between the generalized quadrangle arising from a Hermitian surface H(3, q2) and an elliptic quadric Q−(3, q2) of PG(3, q2). In odd characteristic we determine the possible intersection sizes between H(3, q2) and Q−(3, q2) under the hypothesis that they share the same tangent plane at a common point. When the characteristic is even, we determine the configuration arising from the intersection of H(3, q2) and Q−(3, q2), provided that the generators of H(3, q2) that are tangents with respect to Q−(3, q2) are the extended lines of a symplectic generalized quadrangle W(3, q) embedded in H(3, q2). As a by-product, new infinite families of hyperovals on H(3, q2) are constructed.

In this paper we show that the systems introduced in [Eelbode, Adv. Appl. Clifford Algebr. 17: 635–649, 2007] and [Peña-Peña, Sabadini, Sommen, Complex Anal. Oper. Theory 1: 97–113, 2007] are equivalent, both giving the notion of quaternionic Hermitian monogenic functions. This makes it possible to prove that the free resolution associated to the system is linear in any dimension, and that the first cohomology module is nontrivial, thus generalizing the results in [Peña-Peña, Sabadini, Sommen, Complex Anal. Oper. Theory 1: 97–113, 2007]. Furthermore, exploiting the decomposition of the spinor space into 𝔰𝔭(m)-irreducibles, we find a certain number of “algebraic” compatibility conditions for the system, suggesting that the usual spinor reduction is not applicable.