Hyperbolic complete monotonicity property ($\mathrm{HCM}$) is a way to check if a distribution is a generalized gamma ($\mathrm{GGC}$), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions $E_α, \;α\in (0,2]$, enjoy the $\mathrm{HCM}$ property, and then intervene deeply in the probabilistic context. We prove that, for suitable $α$ and complex numbers $z$, the real and imaginary part of the functions $x\mapsto E_α\big(z x\big)$, are tightly linked to the stable distributions and to the generalized Cauchy kernel.