  Using the worldline SU(2|1) superfield approach, we construct N=4
superconformally invariant actions for the d=1 multiplets (1, 4, 3) and (2, 4,
2). The SU(2|1) superfield framework automatically implies the trigonometric
realization of the superconformal symmetry and the harmonic oscillator term in
the corresponding component actions. We deal with the general N=4
superconformal algebra D(2,1;$\alpha$) and its central-extended $\alpha$=0 and
$\alpha$=-1 psu(1,1|2)$\oplus$su(2) descendants. We capitalize on the
observation that D(2,1;$\alpha$) at $\alpha\neq$0 can be treated as a closure
of its two su(2|1) subalgebras, one of which defines the superisometry of the
SU(2|1) superspace, while the other is related to the first one through the
reflection of $\mu$, the parameter of contraction to the flat N=4, d=1
superspace. This closure property and its $\alpha$=0 analog suggest a simple
criterion for the SU(2|1) invariant actions to be superconformal: they should
be even functions of $\mu$. We find that the superconformal actions of the
multiplet (2, 4, 2) exist only at $\alpha$=-1, 0 and are reduced to a sum of
the free sigma-model type action and the conformal superpotential yielding,
respectively, the oscillator potential $\sim \mu^2$ and the standard conformal
inverse-square potential in the bosonic sector. The sigma-model action in this
case can be constructed only on account of non-zero central charge in the
superalgebra su(1,1|2).
