Is there an algorithm to compute the shortest Hamiltonian path in an directed graph from one point to another in polynomial time?
Here's a small hint.

Let $G$ be a directed graph with unit weights, and $u$ and $v$ be two vertices.  Suppose that I could solve SHP in polynomial time, say, $O(n^c)$.

Now, say that I would like to solve Hamiltonian Path on a directed graph.  Let me spend $O(n^2)$ time to "guess" your vertices $u$ and $v$, and for each of them run your algorithm for SHP.  Then, I would in $O(n^{c+2})$ time have either found a Hamiltonian path in $G$, or conclude that $G$ is not Hamiltonian.

That would mean that P = NP.

Now, this doesn't prove that your problem in NP-complete.  For that, you need to come up with a reduction.  But it does tell you that you should probably not spend too much time looking for a polynomial time algorithm.