[tel]

That definition of abs has merit. In some spaces we are able first to define only an “inner product” between elements p(a, b) and then follow on by naming the length of an element to be sqrt(p(a, a)).

One trick about that inner product is that it need not be perfectly symmetric. To make it work on complex numbers we realize that we have to define it like p(a,b) = a . conj(b) where the . is normal multiplication and the conjugate operation reflects a complex number over the real line.

Now sqrt(p(i+1, i+1)) is sqrt((i+1) . (-i+1)) = sqrt(-i^2 + i - i + 1) = sqrt(2).

I’m skipping over a lot but I wanted to gesture toward where your intuition matches some well known concepts so that you could dive in more deeply. Also wanted to mention the conjugation trick to make your example work!