  We calculated the spectral properties of two related families of
non-Hermitian free-particle quantum chains with $N$-multispin interactions
($N=2,3,\ldots$). The first family have a $Z(N)$ symmetry and are described by
free parafermions. The second one has a $U(1)$ symmetry and are generalizations
of $XX$ quantum chains described by free fermions. The eigenspectra of both
free-particle families are formed by the combination of the same
pseudo-energies. The models have a multicritical point with dynamical critical
exponent $z=1$. The finite-size behavior of their eigenspectra, as well as the
entanglement properties of their ground state wave function, indicate the
models are conformally invariant. The models with open and periodic boundary
conditions show quite distinct physics due to their non-Hermiticity. The models
defined with open boundaries have a single conformal invariant phase while the
$XX$ multispin models show multiple phases with distinct conformal central
charges in the periodic case. The critical exponents of the models are
calculated for $N=3,4,5$ and $6$.
