  We establish quantitative upper and lower bounds for Schr\"odinger operators
with complex potentials that satisfy some weak form of sparsity. Our first
result is a quantitative version of an example, due to S.\ Boegli (Comm. Math.
Phys., 2017, 352, 629-639), of a Schr\"odinger operator with eigenvalues
accumulating to every point of the essential spectrum. The second result shows
that the eigenvalue bounds of Frank (Bull. Lond. Math. Soc., 2011, 43, 745-750
and Trans. Amer. Math. Soc., 2018, 370, 219-240) can be improved for sparse
potentials. The third result generalizes a theorem of Klaus (Ann. Inst. H.
Poincar\'e Sect. A (N.S.), 1983, 38, 7-13) on the characterization of the
essential spectrum to the multidimensional non-selfadjoint case. The fourth
result shows that, in one dimension, the purely imaginary (non-sparse) step
potential has unexpectedly many eigenvalues, comparable to the number of
resonances. Our examples show that several known upper bounds are sharp.
