Question

If ψ(x) is complex would the linear combination Ψ = ψ(x) + ψ*(x) forms a
solution to the time-independent Schrödinger equation?

Answer 1

The time-independent Schrödinger equation

$${\hat H}\psi=E\psi$$

where ${\hat H}$ is a Hamiltonian. It is hermitian operator

$$\hat H^{\dag}=\hat H$$

Hence

$$({\hat H}\psi)^*=(E\psi)^*$$

$${\hat H}\psi^*=E\psi^*$$

Therefore, the linear combination

$$\Psi=\psi+\psi^*$$

is also solution of the time-independent Schrödinger equation.