Claudius writes copiously throughout his life.
Arnaldo Momigliano states that during the reign of Tiberius—which covers the peak of Claudius' literary career—it became impolitic to speak of republican Rome. (Momigliano, Arnaldo [1934] Claudius: the Emperor and His Achievement Trans. W.DHogarth. Cambridge: W. Heffer and Sons.)
The trend among the young historians was to either write about the new empire or obscure antiquarian subjects.
Claudius is the rare scholar who covers both.
Besides the history of Augustus' reign that caused him so much grief, his major works include an Etruscan history and eight volumes on Carthaginian history, as well as an Etruscan Dictionary and a book on dice playing. (Claudius is actually the last person known to have been able to read Etruscan.)
Despite the general avoidance of the Imperatorial era, he has penned a defense of Cicero against the charges of Asinius Gallus.
Modern historians have used this to determine both the nature of his politics and of the aborted chapters of his civil war history.
He has proposed a reform of the Latin alphabet by the addition of three new letters, two of which serve the function of the modern letters W and Y.
He had officially instituted the change during his censorship, but they will not survive his reign.
Claudius has also tried to revive the old custom of putting dots between successive words (Classical Latin is written with no spacing).
Finally, he has written an eight-volume autobiography that Suetonius describes as lacking in taste.
Since Claudius (like most of the members of his dynasty) heavily criticizes his predecessors and relatives in surviving speeches, it is not hard to imagine the nature of Suetonius' charge.
Unfortunately, none of the actual works survive.
They do live on as sources for the surviving histories of the Julio-Claudian dynasty.
Suetonius quotes Claudius' autobiography once, and must have used it as a source numerous times.
Tacitus uses Claudius' own arguments for the orthographical innovations mentioned above, and may have used him for some of the more antiquarian passages in his annals.
Claudius is the source for numerous passages of Pliny's Natural History.
As the author of a treatise on Augustus' religious reforms, Claudius feels himself in a good position to institute some of his own.
He has strong opinions about the proper form for state religion.
He refuses the request of Alexandrian Greeks to dedicate a temple to his divinity, saying that only gods may choose new gods.
He restores lost days to festivals and gets rid of many extraneous celebrations added by Caligula.
He re-institutes old observances and archaic language.
Claudius is concerned with the spread of eastern mysteries within the city and searches for more Roman replacements.
He emphasizes the Eleusinian mysteries which had been practiced by so many during the Republic.
He expels foreign astrologers, and at the same time rehabilitates the old Roman soothsayers (known as haruspices) as a replacement.
Claudius opposes proselytizing in any religion, even in those regions where he allows natives to worship freely.
The results of all these efforts are recognized even by Seneca, who has an ancient Latin god defend Claudius in his satire.