I would definitely not say that sqrt(2) is more controversial than sqrt(-1).

With sqrt(2), we define it into existence, because with all the surrounding bits, it makes sense for it to exist.

With sqrt(-1), we define it into existence, because under the assumption that it exists, you can do some calculations that you can't do otherwise, or you can do them more easily, and the results of those calculations happen to be correct.

sqrt(2) assumes continuity on a single axis. sqrt(-1) assumes an entirely new axis. (Not arguing there is any problem, just putting into words the thoughts I have heard from many students.)

sqrt(2) isn't controversial these days. We might even say it's "obvious", because we're all used to using it from childhood. So you might wonder why maths professors would spend effort proving it in detail. This post is an attempt to remind people that there was a time when sqrt(2) wasn't at all obvious, was controversial, and that it makes sense to see it that way. (And indeed if you go back far enough, -1 was controversial).