  In \cite{S 2017}, Suh gave a non-existence theorem for Hopf real
hypersurfaces in the complex quadric with parallel normal Jacobi operator.
Motivated by this result, in this paper, we introduce some generalized
conditions named $\mathcal C$-parallel or Reeb parallel normal Jacobi
operators. By using such weaker parallelisms of normal Jacobi operator, first
we can assert a non-existence theorem of Hopf real hypersurfaces with $\mathcal
C$-parallel normal Jacobi operator in the complex quadric $Q^{m}$, $m \geq 3$.
Next, we prove that a Hopf real hypersurface has Reeb parallel normal Jacobi
operator if and only if it has an $\mathfrak A$-isotropic singular normal
vector field.
