Can you define Self-adjoint/Hermitian as essentially the same as normal but for real (non-complex) inner product spaces?

Question

Is it possible to have linear operator on complex inner product space that satisfy $T=T^*$?

Answer 1

Yes, as long as you recall that $^*$ means conjugate transponse instead of just the standard transpose. For example, $$ \begin{pmatrix}1&i\\-i&1\end{pmatrix} $$ is equal to its conjugate transpose