Wave amplitude as a complex number?

Question

In section 1-3 An experiment with waves of The Feynman Lectures on Physics (https://www.feynmanlectures.caltech.edu/III_01.html) it says:

"The instantaneous height of the water wave at the detector for the wave from hole 1 can be written as (the real part of) ${h_1}e^{iωt}$, where the “amplitude” $h_1$ is, in general, a complex number."

What does it mean to have a amplitude that is a complex number? I'm used to working with the wave representation of $Ae^{iwt}$ where $A$ is just a constant describing the amplitude, so I'm a bit lost when this amplitude all of a sudden is a complex number.

Answer 1

If $h= |h|e^{i\theta}$ then $he^{i\omega t}= |h|e^{i(\omega t +\theta)}$, so the phase is just shifted.

Answer 2

It's definitely confusing. I think one way to explain this might be that it results from attempting to contain all of the information of the wave in one function. The real component describes the physical aspects of the wave, and then physicists needed to also describe the phase shift. This is baked into the imaginary component.