Can't we teach other intelligent animals "complex" languages? I assume I'm not the first person to think about this, but I find this idea very useful, and if it works it would really be a major breakthrough (IMO, more important than getting to space).

Animals already have their own "kind-of-language", but it is based more on feelings, moods, and whatever the sounds/body-language naturally inspire, and not on abstract thought.
Given that there are primates capable of using sign language, I am left to believe that the barriers preventing animals from learning to speak are not simply their smaller brains, but their inability to pronounce the letters of most human languages and the lack of incentive or a good teacher (human parents are the equivalent language-teacher for the human-baby)

The scenario I though of would be the following (I used chimps as an example, but you can replace it with other animals if you wish):
A group of world-class language-scientists develop a language with words and letters that can be easily pronounced by chimps. The language could even be constructed to be a very "beginner" language, to ease the learning process, as long as it keeps its properties of being able to express various thoughts and ideas.
    A few people then spend a some years to learn and practice this language until they can speak it more or less fluently, and adopt a baby chimp (or more) to raise, like a human baby.

There are, of course, a lot of details left to discuss. The "parent" should probably form a close bond with the chimp, so that it trusts them and pays attention to them. Luck is also a factor, as maybe only some current chimps would be able to pass this experiment.

In summary, my question is: Are there currently any species likely to be able to learn a language specifically made for them? And if yes, is there nobody researching this? I want to talk with my cat!

I am interested in your thoughts about this, and in any useful information you might have regarding this subject.
Many researchers into animal language have presented results of studies as evidence of linguistic abilities in animals. Many of their conclusions have been disputed (Jannedy, Poletto, &amp; Weldon, 1994; Wallman, 1992).
It is now generally accepted that apes can learn to sign and are able to communicate with humans.
The Smithsonian Institution talked about apes communicating and the following are YouTube videos of a gorilla communicating through sign language
A Conversation With Koko
Koko signing with Robin Williams (with translation in subtitles)
References
Jannedy, S., Poletto, R., &amp; Weldon, T. L. (Eds.). (1994). Language files: materials for an introduction to language &amp; linguistics. Ohio State University Press.
Wallman, J. (1992). Aping Language. Cambridge University Press.