To Urinate. Specifically (but not exclusively) in a urinal.

Origin: Certain urinal drains bear an uncanny resemblance to classic comic strip character Ziggy staring up at you. (two beady eye and a giant nose)
If the drain is backed up, or you have a exceptionally robust stream, it is possible to completely submerge the "nose" and therefore "Drown Ziggy"
As nothing good ever happens to Ziggy, it is rather fitting.
"I'll be right back, I need to go Drown Ziggy".
"I drank so much last night that when I took a piss I almost drowned Ziggy."
by Phill the Thrill December 30, 2011
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