You've gone up a degree for each example. The inner product gives you a number and the "cross product" for vectors gives you another vector and is not the same

You've gone up a degree for each example. The inner product gives you a number and the "cross product" for vectors gives you another vector and is not the same thing as the outer product. I think it's also not really correct to compare them like this, because complex numbers give you complex structure, whereas vectors don't. Yes, you get another dimension, but also a rich algebra, Cauchy-Riemann equations, etc.