That's not unique to complex analysis; there are smooth real functions that are nowhere real analytic. It's far easier to come up with (and understand!) examples of functions that are smooth when considered as real maps from R^2 to itself, but nowhere homomorphic when considered as complex functions of a single complex variable, though. Complex conjugation is an obvious example: as the real map (x,y) -> (x,-y), it's linear, so smooth, but only holomorphic at points where 1 = -1, so nowhere. Complex conjugation restricted to the real line does have an obvious holomorphic extension, of course!
That's not unique to complex analysis; there are smooth real functions that are nowhere real analytic.
Did you accidentally get my intended point backwards? My point is that real analysis admits all sorts of fine gradations of counter-examples, and complex analysis does not. In this case I was indeed thinking of a smooth real function that is nowhere analytic. (For non-mathematicians, that would mean a function that can be differentiated any number of times at any point, but which cannot be written as a power series.)
Contrast with complex analysis where continuous at a point and differentiable in a neighborhood of that point implies analytic. Even something as ill-behaved as the point at 0 of the absolute value function is not possible in complex analysis.
