I have a solid understanding of complex numbers, e.g. as 2-dimensional vector space over the reals, as field extension of $\mathbb{R}$ (or more generally as algebraic closure of $\mathbb{R}$) or as matrixes $\left( \begin{array}{rr} a & b \\ -b & a \end{array}\right)$
There is one answer about complex analysis which already helped me: Differences between real and complex analysis?. And i am aware of some major differences like:
- complex numbers can't be ordered
- geometrically you can 'walk around` some points ( $\mathbb{C} \backslash \{0\}$ is connected but $\mathbb{R}\backslash\{0\}$ is not.)
What are some geometric results which are 'obvious' in $\mathbb{R^n}$ but don't hold in $\mathbb{C^n}$ or vice versa?
Two more analytic examples:
1) Hartog's phenomenon. A holomorphic function of more than one variable cannot have an isolated (non-removable) singularity. Also, most domains in $\mathbb{C}^n$, $n>1$ will have a strictly larger hull to which every holomorphic function defined on the domain can be extended. Both are obviously wrong for differentiable functions in real variables (and also for functions of one complex variable, for different reasons).
2) Rigidity of hypersurfaces. Given a hypersurface $M$ (of real codimension $1$) in $\mathbb{C}^n$, $n>1$, satisfying certain conditions, the set of maps holomorphic in a neighbourhood of $M$ which map $M$ into $M$ will be a finite-dimensional Lie-Group. (See for example Ebenfelt, Peter; Lamel, Bernhard; Zaitsev, Dmitri, Degenerate real hypersurfaces in $\mathbb {C}^2$ with few automorphisms, Trans. Am. Math. Soc. 361, No. 6, 3241-3267 (2009). ZBL1174.14038.) It might even happen that there is no such mapping. This does not hold for $\mathcal{C}^\infty$-automorphisms of any real hypersurface, since you can always wiggle around a compact piece of the hypersurface without disturbing anything else.