  We study a quantization via fractional derivative of a nonminimal derivative
coupling cosmological theory, namely, the Fab Four John theory. Its Hamiltonian
version presents the issue of fractional powers in the momenta. That problem is
solved here by the application of the so-called conformable fractional
derivative. This leads to a Wheeler-DeWitt equation of second order, showing
that a Bohm-de Broglie interpretation can be constructed. That combination of
fractional quantization and Bohmian interpretation provides us a new
quantization method, in which the quantum potential is the criterion to say if
a quantum solution is acceptable or not to be further studied. We show that a
wide range of solutions for the scale factor is possible. Among all of those, a
bouncing solution analogous to the perfect fluid cosmology seems to deserve
special attention.
