Nontrivial algorithm for computing a sliding window median
Here's a lower bound from sorting. Given an input set $S$ of length $n$ to be sorted, create an input to your running median problem consisting of $n-1$ copies of a number smaller than the minimum of $S$, then $S$ itself, then $n-1$ copies of a number larger than the maximum of $S$, and set $k=2n-1$. The running medians of this input are the same as the sorted order of $S$.

So in a comparison model of computation, $\Omega(n\log n)$ time is required. Possibly if your inputs are integers and you use integer sorting algorithms you can do better.