You keep saying "complexity". There's a reason I keep writing "K-complexity". The extra K-complexity comes from the extra bits needed to narrow down the results. The whole multiverse does have English-complexity (the conventional meaning of the word, not the measure of how many words it takes to describe something which would just be K-complexity again) greater than the part... but English-complexity is ill-defined.

Look at a word-processing document. A real one, sitting on your hard drive. The program to output all possible documents is very, very simple. The specification of how to get to the exact document you are looking it is the (compressed) size of the document itself. The English-complexity of "the set of all word processing documents" is high, but the K-complexity is low. The English-complexity of "one particular document" is low, but the K-complexity is quite high.

You might say, "Well, I simply tell you to simulate the multiverse, then hand you instructions on how to get to that document", but the instructions will be of a very non-trivial size. I think you intuitively see the instructions as very small, but they are actually huge. Starting with just "Simulate the multiverse" leaves me with, quite literally, a multiverse in hand. Now what? Now how do I find what you are talking about? I'm worse off than when I had nothing at all!

When you have a gigantic set, simply the act of indicating a member within it takes bits. K-complexity measures those bits. English-complexity says you're lowering the complexity. Neither is wrong... it's a definitional matter.