  We construct a Liouville superconformal field theory with eight real
supercharges in four dimensions. The Liouville superfield is an $\mathcal{N}=2$
chiral superfield with sixteen bosonic and sixteen fermionic component fields.
Its lowest component is a log-correlated complex scalar field whose real part
carries a background charge. The theory is non-unitary with a continuous
spectrum of scaling dimensions. We study its quantum dynamics on the
supersymmetric 4-sphere and show that the classical background charge is not
corrected quantum mechanically. We calculate the super-Weyl anomaly
coefficients and find that $c$ vanishes, while $a$ is negative and depends on
the background charge. We derive an integral expression for the correlation
functions of superfield vertex operators in $\mathcal{N}=2$ superspace and
analyze them in the semiclassical approximation by using a quaternionic
formalism for the $\mathcal{N}=2$ superconformal algebra.
