It seems that by identifying CP_{3,h} as the twistor space of M^{4}, one could develop M^{8}H duality to a surprisingly detailed level from the conditions that the dimensional reduction guaranteed by the identification of the twistor spheres takes place and the extensions of rationals associated with the polynomials defining the spacetime surfaces at M^{8} and twistor space sides are the same. The reason is that minimal surface conditions reduce to holomorphy meaning algebraic conditions involving first partial derivatives in analogy with algebraic conditions at M^{8} side but involving no derivatives.
 The simplest identification of twistor spheres is by z_{1}=z_{2} for the complex coordinates of the spheres. One can consider replacing z_{i} by its Möbius transform but by a coordinate change the condition reduces to z_{1}=z_{2}.
 At M^{8} side one has either RE(P)=0 or IM(P)=0 for octonionic polynomial obtained as continuation of a real polynomial P with rational coefficients giving 4 conditions (RE/IM denotes real/imaginary part in quaternionic sense). The condition guarantees that tangent/normal space is associative.
Since quaternion can be decomposed to a sum of two complex numbers: q= z_{1} + Jz_{2} RE(P)=0 correspond to the conditions Re(RE(P))=0 and Im(RE(P))=0. IM(P)=0 in turn reduces to the conditions Re(IM(P))=0 and Im(IM(P))=0.
 The extensions of rationals defined by these polynomial conditions must be the same as at the octonionic side. Also algebraic points must be mapped to algebraic points so that cognitive representations are mapped to cognitive representations. The counterparts of both RE(P)=0 and IM(P)=0 should be satisfied for the polynomials at twistor side defining the same extension of rationals.
 M^{8}H duality must map the complex coordinates z_{11}=Re(RE) and z_{12}=Im(RE) (z_{21}=Re(IM) and z_{22}=Im(IM)) at M^{8} side to complex coordinates u_{i1} and u_{i2} with u_{i1}(0)=0 and u_{i2}(0)=0 for i=1 or i=2, at twistor side.
Roots must be mapped to roots in the same extension of rationals, and no new roots are allowed at the twistor side. Hence the map must be linear: u_{i1}= a_{i}z_{i1}+b_{i}z_{i2} and u_{i2}= c_{i}z_{i1}+d_{i}z_{i2} so that the map for given value of i is characterized by SL(2,Q) matrix (a_{i},b_{i};c_{i},d_{i}).
 These conditions do not yet specify the choices of the coordinates (u_{i1},u_{i2}) at twistor side. At CP_{2} side the complex coordinates would naturally correspond to EguchiHanson complex coordinates (w_{1},w_{2}) determined apart from color SU(3) rotation as a counterpart of SU(3) as subgroup of automorphisms of octonions.
If the base space of the twistor space CP_{3,h} of M^{4} is identified as CP_{2,h}, the hypercomplex counterpart of CP_{2}, the analogs of complex coordinates would be (w_{3},w_{4}) with w_{3} hypercomplex and w_{4} complex. A priori one could select the pair (u_{i1},u_{i2}) as any pair (w_{k(i)},w_{l(i)}), k(i)≠ l(i). These choices should give different kinds of extremals: such as CP_{2} type extremals, string like objects, massless extremals, and their deformations.
String world sheet singularitees and worldline singularities as their lightlike boundaries at the lightlike orbits of partonic 2surfaces are conjectured to characterize preferred extremals as surfaces of H at which there is a transfer of canonical momentum currents between Kähler and volume degrees of freedom so that the extremal is not simultaneously an extremal of both Kähler action and volume term as elsewhere. What could be the counteparts of these surfaces in M^{8}?
 The interpretation of the preimages of these singularities in M^{8} should be number theoretic and related to the identification of quaternionic imaginary units. One must specify two nonparallel octonionic imaginary units e^{1} and e^{2} to determine the third one as their cross product e^{3}=e^{1}× e^{2}. If e^{1} and e^{2} are parallel at a point of octonionic surface, the cross product vanishes and the dimension of the quaternionic tangent/normal space reduces from D=4 to D=2.
 Could string world sheets/partonic 2surfaces be images of 2D surfaces in M^{8} at which this takes place? The parallelity of the tangent/normal vectors defining imaginary units e_{i}, i=1,2 states that the component of e_{2} orthogonal to e_{1} vanishes. This indeed gives 2 conditions in the space of quaternionic units. Effectively the 4D spacetime surface would degenerate into 2D at string world sheets and partonic 2surfacesa as their duals. Note that this condition makes sense in both Euclidian and Minkowskian regions.
 Partonic orbits in turn would correspond surfaces at which the dimension reduces to D=3 by lightlikeness  this condition involves signature in an essential manner  and string world sheets would have 1D boundaries at partonic orbits.
See the chapter TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, M^{8}H Duality, SUSY, and Twistors or the article Twistors in TGD.
