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Isn't it more important to recognise that maths is about adventure, discovery, fun? This is simply an example of an important aspect of in mathematics - you take some concept, abstract it and extend it, and see where it leads. Shouldn't more maths teaching and learning emphasise this?
A nice example that I use in my classroom, far simpler than analytic continuation and one that survives the journey more intact, concerns the index laws. You take the well understood concept that a times a times a... n times is a^n, and uncover the fact that a^m times a^n = a^(m+n). Then generalise the concept of power so you can consider things that "don't make sense" like a^(1/2) or a^(-3). These are not meaningful under the initial view that "power means repeated multiplication", but the extension and subsequent exploration lead to important and very meaningful results. Learning the index laws can be an exercise in rote memorisation, or it can be a wonderful journey of discovery, where seeming "nonsense" becomes clarified and empowering!
Numberphile demonstrate this side of maths, and should absolutely be congratulated.