Are there any examples of hidden Markov models that are not mixture models? Mixture models lend themselves nicely to HMM (hidden Markov model) treatment. Obviously, HMM can be nonmixture or not resolve mixtures when there is only one resolvable input or the superposition of self-similar inputs. What I am asking for is whether or not there are any not so trivial examples of nonmixture HMM data, with the impact relating to an attempt to define HMM as non-trivially different than mixture model analysis, e.g., see https://stats.stackexchange.com/a/249712/99274 $\leftarrow$Let me ask, for example, if the example I gave is exactly correct or not.
Be mindful please that a hidden Markov model can be considered a generalization of a mixture model where the hidden variables (or latent variables), which control the mixture component to be selected for each observation, are related through a Markov process rather than independent of each other.. 
And moreover that In the standard type of hidden Markov model considered here, the state space of the hidden variables is discrete, while the observations themselves can either be discrete (typically generated from a categorical distribution) or continuous (typically from a Gaussian distribution). 
 A: This might deserve to be a comment:
One of the reasons I don't understand your question, or in particular your second sentence, is that I'm not sure unimodal is equivalent with non-mixture. I guess it would be nice to think of those as equivalent if the observed random variables were continuous and in particular Gaussian, but also even in that case, I bet if the locations of each of those Gaussians are close enough, or if the weights are in such a configuration, that one of the conditional distributions of the observations could be unimodal, still. Are we talking about the predictive distribution, or some other conditional distribution? 
And it IS very different because the proportions are changing (this might be more of the answer you're looking for). Assuming a HMM is assuming a model for how the mixing proportions change (the probabilities, the locations, the scales, etc.). You're assuming they are a Markov Chain. A mixture model is a static mixture. A HMM is a dynamic mixture. 
A: HMMs model ordered sequences of observations or time-series $X = \{x_1, x_2,..., x_T\}$, where each $x_i$ can be a discrete or continuous, uni-dimensional or multi-dimensional observation/data sample. The order of the samples, from $t = 1$ to $t=T$ is modeled by the HMM as well (via the transition matrix). 
Mixture models cannot model any dynamics in the data. So HMM can be seen as a Dynamic Mixture Model as noted in this research report for instance that defines the HMMs in the context of Dirichlet mixture models (as their dynamic counterpart). The multimodality does not change anything to this, even when mixtures are not used, HMMs embed data dynamics while mixture models "see" data as unordered. (By the way, as someone commented, multimodal is a very confusing term I find in the context of HMMs...). 
The only cases HMMs are similar to mixture models are if the number of states is 1 or, as said in the comments, if the transition matrix is the identity matrix added to the fact the initial state probability mass function is a vector of 0 with a unique 1, and that simple distributions and not mixtures are assigned to the sates. Otherwise, it is more a "collection" of mixtures. However, knowing that the transition matrix is the identity adds an information about the dynamism of the data, and I would see it as a "weak-equivalence". In the case of a mixture-based HMM, the mixing matrix also impacts making this "equivalence" even more challenged...
In general, a good introduction reading to HMMs is this paper by Rabiner and Juang who popularized their use in the speech processing community.
A: With regrets, I did not want to answer my own question. What I was asking about was something other than mixture modelling. Here is one; Convolution HMM. @GeoMatt22, I had originally thought that "continuous-state HMM with no pre-defined parametric PDFs built in" was the most natural mixture model HMM definition, and did not even know about discrete modelling until it was mentioned here and I started looking into it a bit deeper. The general filter problem may be even more general than convolution as I suspect that its solution may be a general inverse solution, which is very general, indeed. But, the point here is that most people would probably agree that mixture models are distinct from convolution, and from the link, it would seem that there is such a thing as a convolution HMM. Indeed, a highly cited one, 600+. Having said that, I will award +1 to anyone who can (and has) helped with the semantics. What I am trying to do is come up with a short, good definition of HMM. And as @Geomatt22 says, the Wikipedia entry may be a bit 'particular' not to say wonky.
