linear algebra - A question on star-congruence.

We consider $ntimes n$ complex matrices. Let $i_+(A), i_-(A), i_0(A)$ be the number of eigenvalues of $A$ with positive real part, negative real part and pure imaginary. It is well known if two Hermitian matrices $A$ and $B$ are $*-$congruent, then

$$(i_+(A), i_-(A), i_0(A))=(i_+(B), i_-(B), i_0(B)).qquad{(1)}$$
If two general matrices $A$ and $B$ are $*-$congruent, (1) may not hold (can you provide an example?).
Moreover, whether a matrix and its transpose are always $*-$congruent?

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