I should point out this is hardly a thesis Chomsky came up with. I’d say it’s the default position for all of human history.
I thought that channeling a chimp might be considered somewhat farcical rather than taken literally.
Define “complex,” completely and allowing for possible opposing claims. 
As Nova countered Ghomsky 30-some years ago:
Washoe: “Baby in my cup.”
It ain’t Shakespeare. It’s around what my daughter was saying when she was Washoe’s age, which was still better than the poetry my brother thought he could make a living writing.
Had any animal besides a human ever learned anything through language alone? If someone signed a description of an object unknown to Washoe, could Washoe identify that object as the one described, in a different context. ‘A different context’ meaning don’t just describe the object then show it to her. Could she identify it the next day from a group of other unknown objects?
If that is not achieved, I don’t think you can really talk about complex language acquisition, which I would assume means being able to utilize language in some abstract manner, not just utilizing rudimentary vocabulary and rudimentary grammar.
I’m trying to find the relevant passage in Next of Kin which describes how Washoe invented her own sign for ‘bib,’ after insisting that the one she was taught was wrong. So far I can’t find it, but I’ll keep looking.
Just remember that all observations are subjective and interpreted by highly-motivated trainers. There’s a huge Clever Hans factor in all of these experiments.
I’m trying to remember what language facility Bonzo ended up with. That was the precursor experiment.
Did he even know when to go to bed?
Well, one issue is that we can use ‘language’ to mean a few different levels of communication.
If we think it just means using sounds/symbols to communicate information, then language is very widespread: even monkeys and birds have different sounds for ‘danger above’, ‘danger below’ , ‘where’s my mommy?’ and ‘I’m a male; let’s mate’. Or heck, ants have different chemical signals for ‘danger’ and ‘food this way’.
The next level is probably using non-instinctual (new) sounds/symbols, allowing a broader range of information. It seems pretty clear that many apes can in fact learn new, arbitrary symbols and use them to communicate.
Human language, however, is more than that; it uses grammar (including recursion and other abstractions) to communicate far more complex and abstract ideas. It seems that the general (but not absolute) consensus is that apes have not so far shown the capability to use grammar in any meaningful way.