Hyper parameter and complexity in $K$ nearest neighbors
First let's define what a hyper parameter is. In the context of statistical learning, a hyper parameter is a fixed constant that describes the prior distribution for a model's parameters. For example, if $X \sim \text{Binomial}(n,p)$ and $p \sim \text{Beta}(\alpha, \beta)$, then $\alpha, \beta$ would be hyper parameters where $n, p$ would be the "parameters" (I use quotes because $p$ is a random variable now). 

$k$NN is  non-parametric in the sense that you aren't explicitly modeling your data as a function of underlying parameters. $k$ only describes how many neighbors to "learn" from. As $k$ increases the computational complexity can increase; however, it doesn't explicitly control the complexity of the model. An example where model size $k$ acts as a hyper parameter is linear regression. Imagine trying to determine a-priori which $k$ of the $N$ features you would like to select for inclusion in the model.

Given this, I would not say that $k$ describes $k$NN model complexity in the sense that typical Bayesian model selection is and therefore should not be viewed as a hyperparameter.