Did Cleopatra choose eternal beauty?

No.  Cleopatra chose power, not beauty and she already had riches and fame.

We know this from Roman denarius coins from her era that show a woman who was hook-nosed, with thin lips and a pointy chin.  Now some people might find this image of her unattractive, just as they find Mark Antony (her lover who is on the other side of the coin) equally unattractive, but this is racist and superficial.  Does beauty really exist apart from charm, personality and intelligence?  When this coin was made she was getting old -- she already had one child by Julius Caesar and two by Mark Antony -- and she would be dead at the mature age of 39.