The punctured cotangent bundle (excluding the zero section) of the complex projective space of dimension $n$ can be identified with the space:
$(P,Q) | P^2=P, tr(P)=1, PQ+QP = Q, tr(Q) = 0, Q\ne 0$,
Where $P$ and $Q$ are $(n+1)$ dimensional Hermitian matrices. Please see Furutani and Tanaka's article(Proposition 2.3.).
Sketch of the proof: The space of the matrices $P$ is isomorphic as a homogeneous space to $CP^n$, since the action of $U(n+1)$ on it is transitive and the isotropy group of any point is $U(n)$. The conditions on the matrix $Q$ are obtained by taking the derivatives of the conditions on $P$, and the identification of $T^*CP^n$ and $TCP^n$ vuia the Fubini-Study metric.