  The purpose of this article is to study pseudospectral properties of the
one-dimensional Schr\"{o}dinger operator perturbed by a complex steplike
potential. By constructing the resolvent kernel, we show that the
pseudospectrum of this operator is trivial if and only if the imaginary part of
the potential is constant. As a by-product, a new method to obtain a sharp
resolvent estimate is developed, answering a concern of Henry and
Krej\v{c}i\v{r}\'{i}k, and a way to construct an optimal pseudomode is
discovered, answering a concern of Krej\v{c}i\v{r}\'{i}k and Siegl. The
spectrum and the norm of the resolvent of the complex point interaction of the
operator is also studied carefully in this article.
