  The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtained
by replacing the complex numbers with the quaternions, mutatis mutandis, in the
standard construction of the KP hierarchy equations and solutions; it is
equivalent to what is often called the Davey-Stewartson II hierarchy. This
article studies its relationship with the theory of quaternionic holomorphic
2-tori in HP^1 (which are equivalent to conformally immersed 2-tori in S^4).
After describing how the Sato-Segal-Wilson construction of KP solutions
(particularly solutions of finite type) carries over to this quaternionic
setting, we compare three different notions of "spectral curve": the QKP
spectral curve, which arises from an algebra of commuting differential
operators; the (unnormalised) Floquet multiplier spectral curve for the related
Dirac operator; and the curve parameterising Darboux transforms of a conformal
2-torus in S^4 (in the sense of Bohle, Leschke, Pedit and Pinkall). The latter
two are shown to be images of the QKP spectral curve, which need not be smooth.
Moreover, it is a singularisation of this QKP spectral curve, rather than the
normalised Floquet multiplier curve, which determines the classification of
conformally immersed 2-tori of finite spectral genus.
