Shortest Path in Layerwise Complete Graph
Your problem is NP-hard, by reduction from clique. Given a graph $G=(V,E)$ and a number $k$, we construct a graph with $k$ internal layers, each containing $|V|$ vertices. All vertices in a layer are connected to all vertices in the following layer. The weight of an edge is $0$ if the corresponding edge is not in $G$, and $-1$ otherwise. We also add shortcut edges corresponding to edges in $G$ with weight $-1$. The graph $G$ has a $k$-clique iff the shortest path in the new graph has length $-k(k-1)/2$.