I have this old question on probabilistic interpretations of filterbanks I have not completely figured out yet. However some time afterwards I realized it is possible to extend even further to complex numbers and make more general to convolutions and not only filterbanks. Since the complex numbers are an algebraically closed field I was thinking maybe we can derive some more interesting properties for the markov chains given the properties of those numbers.
The mapping I had in mind is: First step the classic $2\times 2$ representation matrix for complex numbers : $$a+bi \to \left[\begin{array}{rr}a&-b\\b&a\end{array}\right]$$ And the second step being the same as the mapping for $\mathbb R$: $$a\to \cases{ \left[\begin{array}{cc}a&0\\0&a\end{array}\right], a >0\\\left[\begin{array}{cc}0&a\\a&0\end{array}\right], a<0}$$ In total giving us ( for each complex scalar ), the block of real non-negative scalars:
$$a+bi \to \left[\begin{array}{cc|cc}a&0&0&b\\0&a&b&0\\\hline b&0&a&0\\0&b&0&a\end{array}\right]$$
can we use some property related to complex numbers to help derive how block-convolution matrices consisting of such blocks ( circulant or Toeplitz for example ) will behave in terms of probability theory.
EDIT As promised to Did, a connection to the tags used signal processing and probability theory:
In signal processing a very common tool/operation is linear & time invariant filtering. If the time discrete signal is represented by a vector with a scalar for each time point, then the convolution can be carried out with multiplication by a circulant or Toeplitz matrix depending on if the convolution is circular or not. Now our construction is remapping scalars used in the filters in these matrices to blocks so we get block-matrices. Now after we have done this we can rescale the matrix by normalizing to get row sum of 1. That way we get a stochastic matrix which has many properties in probability. Now if we transform the scalars in the signal into blocks by the same method, we can start treating the filtering as a probabilistic process described by Markov chains.
Example of circulant and Toeplitz matrices for filter with scalars $[1,2,3]$, designed to work on a vectorized signal of length $6$:
$$ {\bf C} = \left[\begin{array}{cccccc}1&2&3&0&0&0\\ 0&1&2&3&0&0\\0&0&1&2&3&0\\ 0&0&0&1&2&3\\3&0&0&0&1&2\\ 2&3&0&0&0&1\end{array}\right], \hspace{1cm} {\bf T} = \left[\begin{array}{cccccc}1&2&3&0&0&0\\0&1&2&3&0&0\\0&0&1&2&3&0\\0&0&0&1&2&3\\0&0&0&0&1&2\\0&0&0&0&0&1\end{array}\right]$$