  Microscopic determination of alpha-decay half-lives requires structure and
reaction calculations. The structure part is given by the microscopic
distribution of the constituent nucleons, while the relative motion of the
product's decay provides the reaction part. This paper studies the
clusterization of the $0^+$ excited states of $^{44}$Ti arising from the
nucleonic degrees of freedom. The continuum spectra of proton and neutron are
incorporated through the Gamow Shell Model formalism. The alpha-like wave
function is calculated in the weak-coupling approximation. Gaussian effective
interaction in each pair of nucleons is included. The $0^+$ ground and excited
states are compared with experiment and shell model calculations. The wave
function amplitudes are obtained and discriminated by their resonant and
non-resonant contributions. The influence of a four-body truncated basis is
analyzed. Inclusion of the continuum spectra produces a gain in the excited
states of a few hundred keV of energy. One candidate for alpha decay was
identified near the experimental unresolved state $(0,2)^+$ of 6.8 MeV
excitation energy, with a lower limit for the half-life of $\approx 0.8$ ns.
