Projection of a hypercube along a long diagonal. What is this polytope called?

What you are looking for is the vertex-frist projection of a hypercube. Apparently it seems that the 4d analogue has no special name.

You can read this post for more informations.

Edit: I computed the polytope. Read my blog post about it. Here is a teaser:

I got here via A. Schulz's blog entry. As he says, you are looking for the projection of the $(d+1)$-dimensional hypercube (the $(d+1)$-cube for short) along a diagonal. But, actually, any generic projection of the $(d+1)$-cube will give (combinatorially) the same polytope. By "generic" I just mean that the direction of projection is not parallel to any coordinate hyperplane.

A different description is that your polytope is the zonotope obtained as the Minkowski sum of any set of $d+1$ generic vectors in $\mathbb R^d$, where generic means that no $d$ of them lie in a hyperplane.