Assume we have a set of points $\{A,B,C,\dots,Z\}$ in a plane. What is the shortest path which includes all points once, starts at $A$ and ends in $Z$ and $A \ne Z$
I am trying to identify if this has the same complexity as the Traveling Salesman Problem.
Actually I have found that this is the "Messenger Problem" or "Wandering Salesman Problem" which is NP-Hard so it has the same complexity as the TSP.
Yes, they have the same complexity. To reduce to TSP, introduce a dummy node that is adjacent only to $A$ and $Z$.
For a linear reduction in the other direction, you can solve any instance of TSP on $n$ nodes with $n-1$ calls to an oracle for your problem, as follows. Fix any node $A$ and for each $j \not= A$, consider the edge $\{A,j\}$, with length $c_{A,j}$. Solve your problem with $A$ and $j$ as the endpoints, yielding path length $p_j$. An optimal solution to the TSP instance corresponds to $\arg\min\limits_{j\not= A} \{c_{A,j}+p_j\}$.