  In this paper we will prove a shape theorem for the last passage percolation
model on a two dimensional $F$-compound Poisson process, called the Hammersley
model with random weights. We will also provide diffusive upper bounds for
shape fluctuations. Finally we will indicate how these results can be used to
prove existence and coalescence of semi-infinite geodesics in some fixed
direction $\alpha$, following an approach developed by Newman and co-authors,
and applied to the classical Hammersley process by W\"uthrich. These results
will be crucial in the development of an upcoming paper on the relation between
Busemann functions and equilibrium measures in last passage percolation models.
