  We extend the concept of monotonic searchability for self-stabilizing systems
from one to multiple dimensions. A system is self-stabilizing if it can recover
to a legitimate state from any initial illegal state. These kind of systems are
most often used in distributed applications. Monotonic searchability provides
guarantees when searching for nodes while the recovery process is going on.
More precisely, if a search request started at some node $u$ succeeds in
reaching its destination $v$, then all future search requests from $u$ to $v$
succeed as well. Although there already exists a self-stabilizing protocol for
a two-dimensional topology and an universal approach for monotonic
searchability, it is not clear how both of these concepts fit together
effectively. The latter concept even comes with some restrictive assumptions on
messages, which is not the case for our protocol. We propose a simple novel
protocol for a self-stabilizing two-dimensional quadtree that satisfies
monotonic searchability. Our protocol can easily be extended to higher
dimensions and offers routing in $\mathcal O(\log n)$ hops for any search
request.
