I can see some benefits for early students in terms connecting it to the parabola axis of symmetry (-b/2a), and also some good practice with difference of two squares expansion/factorization. Also could be useful in introducing students to the concept of substitution of a variable in an equation.
I wouldn't want to fully replace teaching completing the square though, as that's also very useful for things other than solving quadratics. For example putting an integral like this one into a more amenable form.
\int \frac{1}{x^2 - 2x + 10} \, dx = \int \frac{1}{(x-1)^2 + 3^2} \,dx
BTW. I've used this technique many times in the past. Usually in the context of students first learning to factorize simple integer coeff quadratics in the form x^2 + bx + c with the old "sum = b and product = c" method.
After giving the usual simple exercises like x^2 + 8x + 15\, or x^2 + 5x + 6, better students will sometimes ask, "what if you just cannot find any two such numbers with the required sum and product?". The answer I give is that either no such numbers exist (no real roots) or that they exist but are surds and hence more difficult to find. I give an example like x^2 + 6x +7\, and tell them to try (-3+s) and (-3-s), where "s" is the surd to be determined. I never realized I'd found a "new process".
