It is probably just me, but for the life of me, I'm unable to boil complex analysis down to a few easily digestible principles that are more specific than "holomorphic functions are super nice". Yes, complex analysis is beautiful because of this, but also, how do I organize it into a few coherent intuitions? I particularly refer to the syllabus of a graduate course, consisting typically of the min/max theorem, harmonic functions, MVT, normal families, conformal maps, residue theorem and argument principle, singularities, Rouche's theorem, etc. (for a book, think Taylor's Introduction to Complex Analysis.)
I thought some of these would be that "meromorphic functions are approximately rational functions of some order near singularities", and "holomorphic functions are locally polynomial + error term", but again, the list seems to be unending and unorganized for me.
From a closer perspective, how do I think of it in terms of certain central proof strategies? I see proofs of theorems and sure they make sense. But then, some question statements are just crazy to me (for ex: how can we have a bound on the supremum of the derivative of all functions, holomorphic on the disk, at a point?). The scope of these problems is intimidating and negates any intuition I felt that I've built about the tools I've learnt about.
By the way, I'm saying this as someone who loves measure theory: it is super visual and the proofs are often very intuitive. Moreover, I always think of it as the central object being a "measure space", so that any fact I learn seems to have grown in a natural direction from what I already know(as opposed to, why do I care about normal families or the harmonic equation, as opposed to anything else, on the complex plane?)
I know that this is probably because I've spent way more time with measure theory than complex analysis, but my above feelings impede me from wanting to study more complex. Therefore, I'd love to hear from anyone with more experience what the "essence" of (graduate) complex analysis is for them - in terms of the content, intuitions and proof strategies.
TLDR: For people who enjoy complex analysis, what is the "essence" of complex analysis (particularly, a typical graduate course in it, syllabus mentioned above) for you? In other words, why should a typical graduate course contain the topics it does - why should a professor talk about min/max theorems, harmonic functions, conformal maps and normal families in a single introductory course? Something like "properties of holomorphic functions" is too broad, whereas the above grouping is, of course, too narrow. I’m interested in this from both the perspective of topics involved and the underlying proof strategies.
