Here's an improved version of the puzzle. Again it has
two parts: a cut-and-dried calculation part, and a
more open-ended part. This version makes both parts
easier (I hope).
As preconditions, we are told that a, b, c, and d are
rational numbers. Given that, part 1 is to prove that:
a + b sqrt(2) = c + d sqrt(2) ...
if and only if a=c and b=d.
===============
Again the hint for part 2 is: In what way might part 1
be connected to complex numbers?
There is a definite, nontrivial answer to part 2, namely:
uggc://zngujbeyq.jbysenz.pbz/RkgrafvbaSvryq.ugzy