By all accounts, the historical couple truly were that attached and tempestuous. That is, Cleopatra and Caesar slept together as high-powered political entities. They knew exactly how to use each other.
But Cleopatra and Antony were apparently that much in love and that...dumb about it. A number of historians tut-tut over Roman insularity, used by Augustus to argue that Antony was pitting a "foreign" mistress/power against Rome. They point out, correctly, that Egypt wasn't seen as a different Empire but as part of Rome. Every Roman general was occupying land outside of "Rome."
However, I think the historian's arguments are special pleading. I think any populace would look askance at Antony's behavior: a guy who leaves his wife and family to spend more time than necessary with a royal entity who has already linked herself to Caesar--who is also a problem for Rome. She is, moreover, a canny politician (not the drug-addled character in HBO's Rome) and has her own host of familial complications. And all of this during a Civil War!
In any case, the pair require smoldering passion. Elizabeth and Burton mostly deliver. Critics have accused Taylor of being too shrill and Burton as being too sour. Both criticisms are true.
However, Taylor is far better than most Cleopatras, who far too often are too young, pretty, and pouty. Taylor, at least, comes across as self-possessed. At best, she comes across as tough while being sexy, not sexy in spite of being tough. And Burton exhibits charisma, a great man in diminishment. They are believably romantic and believably dysfunctional.