complex differentiable
Take the ordinary calculus definition of a derivative — the limit of a difference quotient as the increment shrinks to zero — and replace the real number that you nudge by with a complex number. The catch is enormous: in the real line you can only approach a point from the left or the right, but in the complex plane the increment h can shrink to zero from infinitely many directions, along any path you like. Complex differentiability demands that the difference quotient settle on the SAME limit no matter which direction h comes in from. That single requirement turns out to be ferociously strong.
Precisely, a function f defined on an open set is complex differentiable at a point z0 if the limit of (f(z0 + h) - f(z0)) / h, as the complex number h -> 0, exists. The value of this limit is the complex derivative f'(z0). Because h is complex, the quotient is a quotient of complex numbers, and the limit must be the same along every approach — radially, spirally, or otherwise.
Compare this to a real function of two real variables that merely has partial derivatives: those only probe the two coordinate directions and are far weaker. Complex differentiability secretly encodes a pair of linked partial differential equations (the Cauchy–Riemann equations) tying the real and imaginary parts together. This is why a once-complex-differentiable function on an open set is automatically infinitely differentiable and even analytic — a rigidity that has no analogue for real differentiable functions.
f(z) = z^2 is complex differentiable everywhere: (f(z+h) - f(z))/h = (2zh + h^2)/h = 2z + h -> 2z as h -> 0 from any direction. But f(z) = conjugate(z) = x - iy is complex differentiable nowhere: approaching along the real axis gives quotient 1, while approaching along the imaginary axis gives -1, so the limit fails to exist.
Polynomials in z are complex differentiable; conjugation is not.
Being complex differentiable at a single isolated point is a weak and almost useless condition; the power comes from being complex differentiable throughout an open set, which is what 'holomorphic' means. For example f(z) = |z|^2 is complex differentiable only at z = 0 and nowhere else, so it is holomorphic nowhere.