Let the independent processes $M=(M_t, t\ge 0)$ and $X=(X_t, t\ge 0)$ be defined on the probability space $(\Omega, \mathcal F, P)$.
Assume $M$ is a martingale with respect to its natural filtration $\mathcal F^M_t$.
Denote $\mathcal F^X_t$ as the natural filtration of $X$.
Will $M$, due to the independence of the two processes, still be a martingale with respect to the filtration $\mathcal F^M_t\vee \mathcal F^X_\infty$?
If the above is true, and $X$ is a positive increasing and pathwise continuous process with $X_0=0$ a.s., would the process $M(X)=(M_{X_t}, t\ge 0)$ be well defined on the same probability space? In that case would $M(X)$ also be a martingale with respect to the natural filtration of $M(X)$?
I just saw that there is a similar question with answer: Martingale preservation under independent enlargement of filtration
Fix $t>s$ and let ${\cal C}=\{A_M \cap A_X: A_M\in{\cal F}^M_s, A_X\in{\cal F}^X_\infty \}.$
The martingale property for $M$ means that for $A_M\in{\cal F}^M_s$, we have $$\mathbb{E}\left((M_t-M_s){\bf 1}_{A_M}\right)=0.\tag1 $$
Thus, for any set in $\cal C$ the independence of $M$ and $X$ gives
$$\mathbb{E}\left((M_t-M_s){\bf 1}_{A_M \cap A_X}\right) =\mathbb{E}\left((M_t-M_s){\bf 1}_{A_M}{\bf 1}_{A_X}\right)= \mathbb{E}\left((M_t-M_s){\bf 1}_{A_M}\right) \mathbb{E}({\bf 1}_{A_X})=0.$$
Since $\cal C$ is a $\pi$-system that generates ${\cal F}^M_s\vee{\cal F}^X_\infty$, the Monotone Class Theorem says that $$\mathbb{E}\left((M_t-M_s){\bf 1}_{A}\right)=0\tag2 $$ for every $A\in {\cal F}^M_s\vee{\cal F}^X_\infty$. This means that $(M_t)$ is a martingale with respect to the filtration $({\cal F}^M_t\vee{\cal F}^X_\infty).$