Summary: A directed strongly regular graph with parameters \((n, k, t, \lambda, \mu)\) is a \(k\)-regular directed graph with \(n\) vertices satisfying that the number of walks of length 2 from a vertex \(x\) to a vertex \(y\) is \(t\) if \(x = y\), \(\lambda\) if there is an edge directed from \(x\) to \(y\) and \(\mu\) otherwise. If \(\lambda = 0\) and \(\mu = 1\) then we say that it is a mixed Moore graph. It is known that there are unique mixed Moore graphs with parameters \((k^2 + k, k, 1, 0, 1)\), \(k \geq 2\), and \((18, 4, 3, 0, 1)\). We construct a new mixed Moore graph with parameters \((108, 10, 3, 0, 1)\) and also new directed strongly regular graphs with parameters \((36, 10, 5, 2, 3)\) and \((96, 13, 5, 0, 2)\). This new graph on 108 vertices can also be seen as an example of a so called multipartite Moore digraph. Finally we consider the possibility that mixed Moore graphs with other parameters could exist, in particular the first open case which is \((40, 6, 3, 0, 1)\).