  The aim of this article is to enlarge the list of examples of non-autonomous
basins of attraction from our previous paper and at the same time explore some
other properties that they satisfy. For instance, we show the existence of
countably many disjoint Short $\mathbb{C}^k$'s in $\mathbb{C}^k.$ We also
construct a Short $\mathbb{C}^k$ which is not Runge and exhibit yet another
example whose boundary has Hausdorff dimension $2k.$
