Bruce Wayne became old and had hung up the cape and cowl. That is, until high-schooler Terry McGinnis stole the suit and became the new Batman in future Gotham City. At first, he used it to get revenge on his father's murderer. But Bruce saw potential in the boy a deemed him worthy of being the new protector of Gotham City.
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Batman's alter ego is practically his identity which covers him being Batman. His alter ego is Bruce Wayne.
Batman's real name is Bruce Wayne
Alfred was the Butler and Guardian of Bruce Wayne when his parents died- and Bruce Wayne is also known as The Batman.
In the animated series, he never gets to meet Dick Grayson. There are mentions of him and Terry's well aware of his existence, but Bruce and Dick apparently don't talk much anymore.
So people (ie Commissioner Gordon) who meet them both Batman and Bruce Wayne won't realise its the same person.