  We construct the general action for $N=4, d=1$ nonlinear supermultiplet
including the most general interaction terms which depend on the arbitrary
function $h$ obeying the Laplace equation on $S^3$. We find the bosonic field
$B$ which depends on the components of nonlinear supermultiplet and transforms
as a full time derivative under N=4 supersymmetry. The most general interaction
is generated just by a Fayet-Iliopoulos term built from this auxiliary
component.
  Being transformed through a full time derivative under $N=4, d=1$
supersymmetry, this auxiliary component $B$ may be dualized into a fourth
scalar field giving rise to a four dimensional $N=4, d=1$ sigma-model. We
analyzed the geometry in the bosonic sector and find that it is not a
hyper-K\"ahler one. With a particular choice of the target space metric $g$ the
geometry in the bosonic sector coincides with the one which appears in
heterotic $(4,0)$ sigma-model in $d=2$.
