You are right, the "real" wave function for a particle is a complex function $\Psi(\vec{x}, t)$, which follows the time-dependent Schrödinger equation:
$$i\hbar\frac{\partial\Psi(\vec{x}, t)}{\partial t} = \hat{H}\Psi(\vec{x}, t)$$
Where $\hat{H}$ is the hamiltonian operator which gives the total energy of the
system, consisting of the kinetic energy and the potential energy, which is given by the potential energy function $V(\vec{x}, t)$:
$$\hat{H}\Psi(\vec{x}, t) := -\frac{\hbar^2}{2m}\nabla^2\Psi(\vec{x}, t) + V(\vec{x}, t)\Psi(\vec{x}, t)$$
If the potential energy function is truly a function of time as well as position, this equation becomes very difficult to solve. Luckily, however, the potential energy is almost always only dependent on the position and therefore becomes a simpler function $V(\vec{x})$. In that case, the Schrödinger equation can be solved by assuming the wave function $\Psi(\vec{x}, t)$ to be a simple product of a function $\psi(\vec{x})$ solely dependent on the position and a function $f(t)$ that only depends on the time:
$$\Psi(\vec{x}, t) = \psi(\vec{x})f(t)$$
Then we are able to seperate the position and time dependent parts of the equation and from that find that $f(t)$ has to be equal to $e^{-iEt/\hbar}$, where $E$ is some real constant (which can be proven to be the energy of the particle). Furthermore, $\psi(\vec{x})$ must follow the so-called time-independent Schrödinger equation:
$$E\psi(\vec{x}) = \hat{H}\psi(\vec{x})$$
where $E$ is the same constant. It can be proven that $\psi(\vec{x})$ can always taken to be a real function (That is, if you do have a solution to the time-independent Schrödinger equation that isn't always realy, you can always express it as a linear combination of ones that are. It does not mean that every solution necessarily has to be real.). Then we have:
$$\Psi(\vec{x}, t) = \psi(\vec{x})e^{-iEt/\hbar}$$
Remember, that this is only under the assumption, that $\Psi(\vec{x}, t)$ CAN be written as such a product. However, all the other solutions, can then be expressed as linear combinations of these simple time-independent solutions, as Zero already pointed out.
Your question supposes a time-independent wave function that is equal to
$$\psi(x) = Ae^{-\alpha x^2}$$
(This is a one-dimensional function, so I'm leaving the arrow off) Therefore, the time-dependent wave-function you know would be
$$\Psi(x, t) = Ae^{-\alpha x^2}e^{-iEt/\hbar}$$
To find $E$, you can apply the Schrödinger equation:
$$EAe^{-\alpha x^2} = \hat{H}(Ae^{-\alpha x^2})$$
(However, you would need to know the potential energy function you used to derive that wave function)
