I'm confused why people say the phase oscillates in time and the amplitude stays the same (the reason for having complex numbers). But on the other hand, the intensity of a traveling beam does oscillate with time.
It seems that you're getting confused on a few terms. On a wave of constant amplitude $A$, the intensity will indeed oscillate, with an amplitude of $A$. Amplitude refers to the maximum/-minimum point that a wave reaches during its oscillations. As such, if the peaks of oscillation are uniform throughout the wave, it has constant amplitude. To further elaborate on the distinction between oscillating phase and oscillating amplitude, in the following picture, the first and second waves have constant amplitude, and the third has an oscillating amplitude, also known as amplitude modulation.