  Magnetically mediated Cooper pairing is generally regarded as a key to
establish the unified mechanism of unconventional superconductivity. One
crucial evidence is the neutron spin resonance arising in the superconducting
state, which is commonly interpreted as a spin-exciton from collective
particle-hole excitations confined below the superconducting pair-breaking gap
($2\Delta$). Here, on the basis of inelastic neutron scattering measurements on
a quasi-two-dimensional iron-based superconductor KCa$_2$Fe$_4$As$_4$F$_2$, we
have discovered a two-dimensional spin resonant mode with downward dispersions,
a behavior closely resembling the low branch of the hour-glass-type spin
resonance in cuprates. The resonant intensity is predominant by two broad
incommensurate peaks near $Q=$(0.5, 0.5) with a sharp energy peak at $E_R=16$
meV. The overall energy dispersion of the mode exceeds the measured maximum
total gap $\Delta_{\rm tot}=|\Delta_k|+|\Delta_{k+Q}|$. These experimental
results deeply challenge the conventional understanding of the resonance modes
as magnetic excitons regardless of underlining pairing symmetry schemes, and it
also points out that when the iron-based superconductivity becomes very
quasi-two-dimensional, the electronic behaviors are similar to those in
cuprates.
