The primary aim of this study is to analyze the chaotic dynamics of a conformable maturity structured cell partial differential equation of order $z\in (0,1)$, which extends the classical Lasota equation. To examine the chaotic behavior of our model's solution, we initially extend certain criteria of linear chaos to conformable calculus. This extension is crucial because the solution of our model does not generate a classical semigroup but rather a $c_0$-$z$-semigroup. For the velocity term of our model, $B(\mathfrak{w})=\mu \mathfrak{w}$, where $\mu\in\mathbb{C}$, and the term source $g(\mathfrak{w}, \vartheta(\mathsf{r}, \mathfrak{w}))$, we utilize spectral properties of the $z$-infinitesimal generator to demonstrate chaotic behavior in the space $C(\mathrm{J}_0, \mathbb{C})$, $\mathrm{J}_0:=[0,+\infty)$. Furthermore, by employing conformable admissible weight functions and setting $B(\mathfrak{w})=1$, we establish chaos in the solution $z$-semigroup, this time within the space $C_{0}(\mathrm{J}_0, \mathbb{C})$.
