Determining the time dependence of the Schrödinger Equation (separation of variables)

In this video I will determine the time dependence of the Schrödinger Equation by separation of variables. Later, I will talk about the significance of the results.

In the next video we will (finally) solve the time independent Schrödinger equations for some simple potential.

My name is Nick Heumann, I am a recently graduated physicist. I love to teach physics, so I decided to give YouTube a try. English is not my first language, but I hope that you can understand me well enough regardless.
▬ Contents of this video ▬▬▬▬▬▬▬▬▬▬
00:00 Start
00:30 Defining the separated functions
03:50 Separating the Variables
07:20 Solving the time dependent equation
10:10 Concept 1: Stationary States
12:25 Concept 2: Definite Energy
17:20 Completeness of the solutions

[yeast]

It looks like you're going through Griffith's book, so maybe you can help me? On page 12 he shows that for any solution of the Schrodinger equation, normalization is preserved over time. When doing the separation of variables, there doesn't seem to be any constraints on E being strictly real, meaning that you could take E to be of the from a+bi and still have a solution to the Schrodinger equation, right? Page 12 would then suggest that it remains normalized over time too. But one can clearly check that when E has non-zero complex part, we find that that it does not remain normalized over time due to an exponential term of e^(2bt/h) when we integrate the modulus squared. What exactly am I missing here?