  In the present paper, we are concerned with the general localized solutions
for the complex short pulse equation including soliton, breather and rogue wave
solutions. With the aid of a generalized Darboux transformation, we construct
the $N$-bright soliton solution in a compact determinant form, then the
$N$-breather solution including the Akhmediev breather and a general higher
order rogue wave solution. The first- and second-order rogue wave solutions are
given explicitly and illustrated by graphs. The asymptotic analysis is
performed rigorously for both the $N$-soliton and the $N$-breather solutions.
All three forms of the localized solutions admit either smoothed-, cusped- or
looped-type ones for the CSP equation depending on the parameters. It is noted
that, due to the reciprocal (hodograph) transformation, the rogue wave solution
to the CSP equation is different from the one to the nonlinear Schr\"odinger
(NLS) equation, which could be a cusponed- or a looped one.
