Consider the [Metropolis-Hastings algorithm][1] which is an [MCMC][2] method, i.e., a general purpose Monte Carlo method for producing samples from a given probability distribution. The method works by generating a Markov chain from a given *proposal Markov chain* as follows.  A proposal move is computed according to the proposal Markov chain, and then accepted with a probability that ensures the Metropolized chain preserves the given probability distribution.  

This Metropolized chain produced by the Metropolis-Hasting algorithm is a "surprising example of a Markov chain" because the acceptance probability at every step of the chain depends on both the current state of the chain and the proposed state.  However, the surprise wears off a bit once one realizes that the next state of the chain does not necessarily coincide with the proposed move, and that the next state of the chain could in fact be the current state of the chain if the proposed move is rejected.


  [1]: https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm
  [2]: https://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo