  We give estimates for the eigenvalues of multi-form modified Dirac operators
which are constructed from a standard Dirac operator with the addition of a
Clifford algebra element associated to a multi-degree form. In particular such
estimates are presented for modified Dirac operators with a $k$-degree form
$0\leq k\leq 4$, those modified with multi-degree $(0,k)$-form $0\leq k\leq 3$
and the horizon Dirac operators which are modified with a multi-degree
$(1,2,4)$-form. In particular, we give the necessary geometric conditions for
such operators to admit zero modes as well as those for the zero modes to be
parallel with a respect to a suitable connection. We also demonstrate that
manifolds which admit such parallel spinors are associated with twisted
covariant form hierarchies which generalize the conformal Killing-Yano forms.
