If we assume - and we probably should - that "complete sentence" means a complete statement or question which can be coherently understood by the reader in the sense the writer intended and count punctuation as part of the sentence, then one need look no further than Oscar Wilde. At the time he was in exile in Paris and sent a telegram to his publisher in London in order to ascertain the current success (or otherwise) of his latest book. Being somewhat lacking in funds, he endeavoured to keep the query as short as possible. He sent "?"; yes, just a single question mark. His publisher, entering into the spirit of the game, replied - signalling, no doubt, the surprising response to the work - a "!". These surely rate as the shortest possible English sentences, if they were sent by Oscar Wilde to his British publisher.
Unfortunately, this story is also attributed (with greater certainty) to an exchange between Victor Hugo and his publisher, which would make them the shortest sentences in the French language. This exchange is said to concern the publication of Les Miserables and the reply expressed the delight that the first print run of 7,000 copies sold out in 24 hours. This author first heard the story about Hugo rather than Wilde and it wasn't until researching this article that the Wilde credit became apparent.
However, let's be honest: the query wasn't really in French; it was a question that would be understood in any language that uses the "?" glyph for the interrogative voice (i.e., most European languages). So it could be said to be in English, even if it originated with a Frenchman.
Like all "shortest sentences", this example relies heavily in the context of the conversation - and implicit information communicated between the parties prior to the conversation - for the meaning to be clear. Victor Hugo's publisher (or Oscar Wilde's) was well aware that the writer was out of town and would wish to know how well the new book was doing. Both were aware that there was some question about how successful the enterprise would be. In this context the short sentences make complete sense.
Arguments will, of course, persist about the legitimacy of an English sentence containing no English words whatsoever, but it is undeniably true that this exchange is emblematic of the creativity inspired by quests for very short language examples. And, all other arguments aside, it is a very amusing story and its multiple attribution makes in a wonderful subject for discussion.
[AE-TG3] Answers To Everything; "Question #48940"; http://www.funtrivia.com/askft/Question48940.html
[DS-OW2] The Daily Star; "Telegrams going the way of Pony Express"; http://www.thedailystar.com/opinion/edits/2006/02/ed0208.html
[NP-OW1] Natalya Predtechenskaya; "Telegram Sam"; http://archive.1september.ru/eng/2000/no15_2.htm
[SO-TG1] SOWMYA; "Shortest telegram"; http://shallowthgts.blogspot.com/2006/02/shortest-telegram.html
[TB-TG2] RebabBlog; "El ultimo telerama"; http://www.becariosbarrie.org/blog/2006/02/03/el-ultimo-telegrama