  We consider a normal operator $T$ on a Hilbert space $H$. Under various
assumptions on the spectrum of $T$, we give bounds for the spectrum of $T+A$
where $A$ is $T$-bounded with relative bound less than 1 but we do not assume
that $A$ is symmetric or normal. If the imaginary part of the spectrum of $T$
is bounded, then the spectrum of $T+A$ is contained in the region between
certain hyperbolas whose asymptotic slope depends on the $T$-bound of $A$. If
the spectrum of $T$ is contained in a bisector, then the spectrum of $T+A$ is
contained in the area between certain rotated hyperbola. The case of infinite
gaps in the spectrum of $T$ is studied. Moreover, we prove a stability result
for the essential spectrum of $T+A$. If $A$ is even $p$-subordinate to $T$,
then we obtain stronger results for the localisation of the spectrum of $T+A$.
