  A matchstick graph is a plane graph with edges drawn as unit distance line
segments. This class of graphs was introduced by Harborth who conjectured that
a matchstick graph on $n$ vertices can have at most $\lfloor 3n - \sqrt{12n -
3}\rfloor$ edges. Recently his conjecture was settled by Lavoll\'ee and
Swanepoel. In this paper we consider $1$-planar unit distance graphs. We say
that a graph is a $1$-planar unit distance graph if it can be drawn in the
plane such that all edges are drawn as unit distance line segments while each
of them are involved in at most one crossing. We show that such graphs on $n$
vertices can have at most $3n-\sqrt[4]{n}/10$ edges.
