Fourier transform of even/odd parts of a complex signal
Defining
$$ x_e(t) = \frac{x(t) + x(-t)}{2} $$
$$ x_o(t) = \frac{x(t) - x(-t)}{2} $$
We can take the Fourier transform of each:
$$
FT\{x_e(t)\} = \int_{-\infty}^{+\infty} x_e(t) (\cos(2\pi\omega t) - j \sin(2\pi\omega t)) dt
$$
$$
FT\{x_o(t)\} = \int_{-\infty}^{+\infty} x_o(t) (\cos(2\pi\omega t) - j \sin(2\pi\omega t)) dt
$$

So if $x_e(t) : \mathbb{R} \mapsto \mathbb{R} $ then the $\sin$ term disappears and we are left with just a real-valued function. 

So if $x_o(t) : \mathbb{R} \mapsto \mathbb{R} $ then the $\cos$ term disappears and we are left with a purely imaginary-valued function.

Now, if instead $x_e(t), x_o(t) : \mathbb{R} \mapsto \mathbb{C} $ then the term cancellations still happen, but we are still left with a complex value for each integral because both functions are now complex-valued.

As a result, the right-hand sides of your quoted equations do not hold in general.