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The two-dimensional topological A model associates to symplectic and contact manifolds rich algebraic invariants such as the Fukaya category. When the symplectic structure is the real part of a complex symplectic structure, these invariants are expected to radically simplify, by ideas of Joyce and Donaldson-Thomas. However, the complex symplectic structure simplifies some aspects while enhancing others. In particular, by complexifying the 2d A model to the 3D A-model, which is governed by the Fueter map PDE, one should be able to associate a 2-categorical invariant to a hyperkahler manifold. Trying to puzzle out this idea leads to interesting problems in PDE and differential geometry, some of which have solutions. I will review this story and some recent developments. This based on joint work with Aleksander Doan, which proves a basic adiabatic limit result that can be viewed as a complex analog of Floer's computation of the Floer homology of a small Hamiltonian. If time permits, I will speculate about the geometry of `quaternionic Weinstein domains'.

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