There are also different variations to this classic suit. There is, of course, the very first appearance (that happens to have a double-breasted tuxedo jacket with a shawl lapel - I've only ever seen them with peak lapels, but I'm sure they exist), which was also used for the Batman '66 comics, albeit with purple gloves instead of white, and purple spats...
There's also the variant seen in "The Killing Joke", with a pleated shirt instead of a plain front, and is also single breasted instead of double breasted...
The "Injustice: Gods Among Us" variant, with a mini-check grey/black suit with a purple waistcoat, black bowtie with white dots, white pinstripe shirt and purple gloves...
There's the variant seen in "Batman: Arkham Origins", which just had the dome and cape over his usual costume...
And then there's the variants seen in the recent New 52 storyline "Zero Year", which featured the Red Hood in a cape-less dome, donning a black suit jacket, tie, trousers, white shirt and gloves...
And in the finale donning a black tailcoat, grey waistcoat, white shirt (pleated with wing collar), black bowtie, white socks and black shoes, with the cape draped around his shoulders (but not attached to the helmet as in previous interpretations), with a hood.
I'm sure there are more, but this is what I could come up with