Question 1: What do holomorphic functions 'look like'? Not really sure what is meant by this, but I heard this was asked this in an interview for graduate school admissions.
What I have in mind is that
A. holomorphic functions are conformal, infinitely complex differentiable or something (not sure of the precise wording, but i guess it's like 'complex smooth'), equivalent to complex analytic (therefore we can just drop the qualifier 'complex') and have harmonic real and imaginary parts (i didn't learn in elementary complex analysis, but in miranda's book 'harmonic' is defined precisely as this. in elementary complex analysis, i learned holomorphic implies harmonic parts). Not really sure how to graph any of these, but the conformal thing is on wiki.
Here's the picture:
B. when you're asked what continuous functions $f: \mathbb R \to \mathbb R$ 'look like', intuitively/heuristically, you can draw a part of it on a paper without lifting your writing utensil from the paper. so you probably just do something that looks like sin/cos or a polynomial. (i remember in calc 1, the 1st continuous functions we're introduced to are these smooth functions sin/cos and polynomials.) I think this is kind of what the question means.
C. from the very definition itself, i just think to draw some circles like $g: \mathbb C \to \mathbb C$ holomorphic at $z=w$ means complex differentiable in some disc/disk around $z=w$. but i guess i'm just making drawings on its domain rather drawing the function itself.
Question 2: what books answer this?
Question 3: In relation to (B) above I actually find this question kind of weird. I mean does it make sense to ask what differentiable real functions $f: \mathbb R \to \mathbb R$ 'look like' ? twice differentiable functions $f: \mathbb R \to \mathbb R$ 'look like'? thrice? smooth functions $f: \mathbb R \to \mathbb R$? real-analytic functions?
Question 4: What do poles look like? what do poles 'look like'? Please provide references.
Update: Based on the comments below, this version of the Maximum modulus principle seems to be relevant:
Let $D \subseteq \mathbb C$ be bounded, non-empty and open. Suppose $f: \overline D \to \mathbb C$ is continuous on $\overline D$ and holomorphic on $D$. Then $|f|$ attains a maximum at some point in boundary of $D$ (i guess topological boundary, and i guess equal to $\overline D \setminus D$).
The comments Severin Schraven include:
It makes quite a lot of sense to ask this question in my opinion. The point is that holomorphic functions have a lot of structure and force them to have certain features. (...) The maximum principle is a very strong property. General continuous functions can be extremely wild!
A comment of Moishe Kohan:
Strictly speaking, the question is meaningless unless the meaning of "looks like" is specified. For this reason, no self-respecting textbook will discuss this, nor an interviewer at a grad school interview. They may ask: "What local properties of holomorphic functions do you know?" This would be a reasonable "long list" question.
God bless you, Moishe Kohan.