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In Closing Time The Doctor shows off the fact that he can 'speak baby' with Stormaggedon. In previous episodes we see that the TARDIS lets anyone who's been in it can speak as though they've got a universal translator.
Amy Pond never seems to speak to her child, though neither does the Doctor at that point in time so is this just a natural talent of the doctor, or is it an upgrade the TARDIS provides?
Knowing the Doctor's quirky sense of humor, it may indeed be the TARDIS making a translation, but it could also be the Doctor using 'Stormaggedon' to make a point in the conversation.
Given what we know about human development, the child should still lack the cognitive abilities to formulate such complex thoughts.
In any conversation with the Doctor, it is prudent to remember Rule Number 1: The Doctor lies.
The TARDIS translates for everyone (as Rose explains in "The Christmas Invasion", when the Doctor regains consciousness and they can understand the Sycorax). So if Amy can't understand her baby, and Craig can't understand Stormageddon, when the Doctor can, it follows that the Doctor knows Baby, and the TARDIS isn't translating for him on this occasion.
Bluntly, there's no specific evidence that the Doctor actually speaks baby (or horse for that matter), even if such a thing was possible. The claims he makes may be part of his own fantasies, outright lies or, as seems more likely used as misdirection to keep the person he's speaking to off-balance.
There are no indications by the babies he speaks to that they've understood him. Alfie, for example shows no particular interest in the Doctor, nor uses any visual cues (blinking, smiling, etc) to suggest that he's comprehending anything that's being said. In that instance, it seems far more likely that the Doctor is simply having a bit of fun with the new father.
Although there's no canon evidence, it also seems reasonable to assume that the Doctor is also trying to alleviate his own distressing memories (e.g. at having lost his own children in the Time War) by constantly making jokes.
Certainly when the Doctor make a similar claim in "A Town Called Mercy" that he speak horse this is the first time in many occasions where he's ridden horses, that he says that he can actually speak to them. The fact that he does so in front of a bible-thumping preacher is especially telling.
PREACHER: He's called Joshua. It's from the Bible. It means the Deliverer.
DOCTOR: No, he isn't. I speak horse. He's called Susan, and he wants you to respect his life choices.
