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Let H be the space of quaternions, with its standard hypercomplex structure.Let R(D) be the module of regular functions on D. For every unitary vector p inS^2, R(D) contains the space of holomorphic functions w.r.t. the complexstructure J_p induced by p. We prove the existence, on any bounded domain D, ofregular functions that are not J_p-holomorphic for any p. Our starting point isa result of Chen and Li concerning maps between hyperkaehler manifolds, where asimilar result is obtained for a less restricted class of quaternionic maps. Wegive a criterion, based on the energy-minimizing property of holomorphic maps,that distinguishes J_p-holomorphic functions among regular functions.

Holomorphic functions and regular quaternionic functions on the hyperk\"ahler space H