After I got selected to the university, one day I visited my grandma. She asked me what my favorite subject was. I said, "My favorite subject is Complex Analysis". Then she said "what is this, Complex Analysis?".
She never studied Complex Analysis and she is not educated. But she has studied mathematics at her primary school. So I need to explain her what Complex Analysis is, roughly. Does anyone know eye-opening words? Perhaps an analogy or a metaphor would help.
I reckon your grandma does not have the notion of function, but she will maybe remember plane geometry. So tell her that complex analysis is doing plane geometry in a special computational way. You write down the equation $z^6=1$, and in the whirl of a moment a regular hexagon inscribed in the unit circle is generated. Depending on the degree of sophistication you can show her the formula for the area of a triangle with vertices $z_1$, $z_2$, $z_3$. Tell her also that using these "complex numbers" helps in describing and understanding circular motion.
In my experience people who ask these kinds of questions are not usually interested in any of the mathematical details. I would say that while familiar mathematics is done with ordinary numbers, positive, negative, and zero, it transpires that these ordinary numbers are incomplete, and are part of a larger system that includes more numbers. By translating questions into this larger system, certain aspects of mathematical problems become clear that were obscure when we were not looking at the whole picture.
This does not get to the point of what "analysis" means, but I think it gets to the point of what complex analysis is about, and it is probably enough for your grandmother. If she asks for further details (which I imagine is unlikely) you might be able to explain that there is a series which looks as if it should converge to $\frac1{1+x^2}$ for all $x$, but it doesn't; why not? And the answer is, although that function is defined for all real numbers, in the complex plane it is badly-behaved at $x=\pm i$. If you get this far she might ask why anyone cares whether the series converges, but the answer to that is easy: when they work, these series are an essential method in obtaining solutions to engineering and physics problems involving differential equations, and you need to know whether the series will actually produce the correct answer.
To give intuition, complex analysis is often described as a useful tool to solve real problems using imaginary numbers, resembling to across a neighborhood country with to reach a location faster than going around the concave boundary. We are just getting a permission to use roads of the country of imaginary numbers. Complex numbers can be thought as a generalization just like how we append negative numbers and real numbers in order to natural numbers to achieve more flexibility.
