A triholomorphic map u between hyperKahler manifolds solves the "quaternion del-bar" equation du=I du i + J du j + K du k. Such a map turns out, under suitable assumptions, to be stationary harmonic. We focus on compactness issues regarding the quantization of the Dirichlet energy and the structure of the blow-up set. We can relax the assumptions on the manifolds, in particular we can take the domain to be merely "almost hyper-Hermitian": this more general setting leads to the weaker notion of"almost-stationarity", without however affecting our compactness results and it leads e.g. to gauge-theoretic applications. This is a joint work with G. Tian (Princeton).