  This paper gives a new perspective on how to solve the second-order linear
differential equation written in normal form. Extending the argument of the
potential to a complex number leads to solving exactly the Schr\"odinger
equation when the potential is complex using the factorization method. This
method leads to solving two Riccati nonlinear equations and by constructing the
only possible superpotential, the factorization method gives the eigenvalues
and eigenfunctions in closed form for potentials satisfying the shape
invariance property. Extending the potential to the complex argument has led to
discovering new exactly solvable ones. In this first part, the basic
superpotentials are divided into different groups, each group contains the
superpotentials that share common terms. All of the already known solvable real
potentials will fall into this category and are derived as special cases. This
set of exactly solvable complexified potentials has already uncovered some of
the properties of quantum mechanics, like the tunneling effect through the
forbidden region, happening with high probabilities between multiwells, bound
states in the continuum (BIC), and other properties. These results have
potential applications in all fields of sciences, from physics, chemistry,
biology, etc., where the eigenvalue problem plays an important role.
