I'm sorry without some citations this is still handwavy bollocks.

Energy does have a conservation law. Entropy does not. Not does information, complexity etc.

I majored in phyiscs, I'm familiar with these terms.

Grand parent post made no indication they were talking about essential complexity when they made their bold assertion about how it's conserved like energy.

I'm also not aware of any formal or precise way of measuring "essential complexity". Not in the way we do for energy or entropy.