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We fully describe the horofunction boundary $\partial_h L_2$ with the wordmetric associated with the generating set $\{t,at\}$ (i.e the metric arising inthe Diestel-Leader graph $\text{DL}(2,2)$). The visual boundary$\partial_\infty L_2$ with this metric is a subset of $\partial_h L_2$.Although $\partial_\infty L_2$ does not embed continuously in $\partial_h L_2$,it naturally splits into two subspaces, each of which is a punctured Cantor setand does embed continuously. The height function on $\text{DL}(2,2)$ provides anatural stratification of $\partial_h L_2$, in which countably-manynon-Busemann points interpolate between the two halves of $\partial_\inftyL_2$. Furthermore, the height function and its negation are themselvesnon-Busemann horofunctions in $\partial_h L_2$ and are global fixed points ofthe action of $L_2$.

The horofunction boundary of the lamplighter group $L_2$ with the Diestel-Leader metric