https://news.ycombinator.com/item?id=26329883 - like this. AIU the article disproves a specific theory that threw away the imaginary part of wave function or something like that, so the result is kinda expected, but it doesn't mean that complex numbers are indispensable.
You can say that about anything, like real quantum mechanics isn't just real numbers, but also functions, equations, operators and a lot of stuff like that.
All of those are part of Peano and ZFC axiomatic systems, which were thought to be enough to explain all of physics (albeit not easily). Matrices and imaginary numbers are not part of those formal systems.
Just ask yourself. If you are right, why didn't Schrodinger or the researches that are publishing this study just think of your idea of using matrices?