Here I found a summary about two-dimensional plane waves.
I have a maybe naive question.
The right-hand side of the first figure shows a plot of $\Re(\exp(i\theta))$ as a function of $(x,y)$ for fixed $t$ and the lines are are lines of constant phase (highs and lows).
(1) Wouldn't it be more illustrative/ intuitive to plot this with a third axis, i.e. giving each point $(x,y)$ its value $\Re(\exp(i\theta))$? Since then we can actually see the "mountain landscape" at ´fixed time $t$?
(2) Then, when evolving $t$. the "mountain landscape" would travel normal to the lines of constant phase, i.e. the whole mountain landscape would move normal to these lines?
I do agree with you that we can use a 3D plot to visualize what you mentioned. Your notes in the pdf file used 2D plots, probably because they are easier to draw.