geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
A function $f$ between complex manifolds is holomorphic if it is complex-differentiable, or equivalently complex-analytic. One may also see definitions referring to the intermediate notions of continuous differentiability or infinite differentiability.
The theorem that every continuously complex-differentiable function is analytic is soft and due to Augustin Cauchy; the theorem that every complex-differentiable function is continuously differentiable is hard and due essentially to Édouard Goursat (which may be seen as filling in a gap in Cauchy’s proof). See Cauchy integral formula and Goursat theorem.
On infinite-dimensional manifolds, we have several notions of holomorphic function; see Wikipeda.