As a non-mathematican, I found that trying to introduce C as a closure of R (i. e. analytically in author's terms) invariably triggers confusion and "hey, why do mathematicians keep changing rules on the fly, they just told me square of minus one doesn't exist". And in terms of practical applications it doesn't seem particularly useful on the first glance (who cares about solving cubics algebraically? The formula is too unwieldy anyway.) Most applications tends to start in the coordinate view and go from there. And it does introduce a nasty sharp edge to cut oneself on (i vs -i), but then for instance physics is full of such edges: direction of pseudo-vectors, sign of voltage on loads sources, holes in dimensional analysis (VA vs W, Ohm/square), the list could go on. And nobody really care.
> "hey, why do mathematicians keep changing rules on the fly, they just told me square of minus one doesn't exist
Mathematicians aren’t chasing numerical solutions, they’re chasing structure. ℂ isn’t just about solving cubics, it’s about eliminating holes in algebra so the theory behaves uniformly and is easier to build upon.
And as for "changing rules" they haven't changed, they have broadened the field (literally) over which the old rules applied in a clever way to remove a restriction.
