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We prove that a Hausdorff locally compact semitopological bicyclic semigroupwith adjoined zero $\mathscr{C}^0$ is either compact or discrete. Also we showthat the similar statement holds for a locally compact semitopological bicyclicsemigroup with an adjoined compact ideal and construct an example whichwitnesses that a counterpart of the statements does not hold when$\mathscr{C}^0$ is a \v{C}ech-complete metrizable topological inversesemigroup.

On the dichotomy of a locally compact semitopological bicyclic monoid with adjoined zero