Suppose it has finite state space $S$, and $\lim\limits_{n\to \infty}p_{ij}^{(n)}=0$ for all $i,j\in S$. But guess is there isn't, since for a finite transition matrix, it is unlikely to have $P^n\to0$ for each of its entry. I have tried some inequalities and summed over $n$ or state space, trying to find contradiction by recurrence theorem, but got nothing.
Any hint would be appreciated!
The sum of each column in transition matrix has to equal 1 (consider a perfectly located state in the beginning - after the matrix multiplication the probability vector will represent the corresponding matrix column). In your case it would be equal to zero.
You can also think of it this way - you would converge to a state where the imaginary particle is nowhere. It is always "somewhere".