Let $(X_n)_{n \in \mathbb{N}}$ be a (time-homogeneous) Markov chain with countable state space $I$ and transition matrix $P$ and assume $X_0=i \in I$. Define the stopping times $T_0:=\inf\{n>0:X_n \neq X_0\}$ and for $k\geq 1$, $T_k:=\inf\{n>T_{k-1}:X_n \neq X_l \text{ for } l = 0,\dots,T_{k-1}\}$. Assume all stopping times are finite $\mathbb{P}$-a.s. (so you can apply strong Markov property).
Is the embedded process $Y_n:=X_{T_{n}}$ a Markov chain?
So the process $Y_n$ is essentially the process $X_n$ with the removal of any states that it has already visited. Now the future of $Y$ is not really independent of the past, given the present. But its a bit weird because the stopping times take care of that....I am really unsure.