  Using the N=4, 1D harmonic superspace approach, we construct a new type of
N=4 supersymmetric mechanics involving 4n-dimensional Quaternion-K\"ahler (QK)
1D sigma models as the bosonic core. The basic ingredients of our construction
are {\it local} N=4, 1D supersymmetry realized by the appropriate
transformations in 1D harmonic superspace, the general N=4, 1D superfield
vielbein and a set of 2(n+1) analytic "matter" superfields representing (n+1)
off-shell supermultiplets (4, 4, 0). Both superfield and component actions are
given for the simplest QK models with the manifolds \mathbb{H}H^n =
Sp(1,n)/[Sp(1) x Sp(n)] and \mathbb{H}P^n = Sp(1+n)/[Sp(1) x Sp(n)] as the
bosonic targets. For the general case the relevant superfield action and
constraints on the (4, 4, 0) "matter" superfields are presented. Further
generalizations are briefly discussed.
