  We study eigenvalues of non-self-adjoint Schr\"odinger operators on
non-trapping asymptotically conic manifolds of dimension $n\ge 3$.
Specifically, we are concerned with the following two types of estimates. The
first one deals with Keller type bounds on individual eigenvalues of the
Schr\"odinger operator with a complex potential in terms of the $L^p$-norm of
the potential, while the second one is a Lieb-Thirring type bound controlling
sums of powers of eigenvalues in terms of the $L^p$-norm of the potential. We
extend the results of Frank (2011), Frank-Sabin (2017), and Frank-Simon (2017)
on the Keller and Lieb-Thirring type bounds from the case of Euclidean spaces
to that of non-trapping asymptotically conic manifolds. In particular, our
results are valid for the operator $\Delta_g+V$ on $\mathbb{R}^n$ with $g$
being a non-trapping compactly supported (or suitably short range) perturbation
of the Euclidean metric and $V\in L^p$ complex valued.
