While moving my laptop the other day, I ended up mashing the keyboard a little, and by pure chance managed to do a google search for `i!`

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Curiously, Google's calculator dutifully informed me that $i!$ was, in fact, $0.498015668 - 0.154949828i$.

Why is this?

I know what a factorial is, so what does it actually mean to take the factorial of a complex number? Also, are those parts of the complex answer rational or irrational? Do complex factorials give rise to any interesting geometric shapes/curves on the complex plane?