Childhood memory: I learned that i is formally defined to be the square root of -1. Well, I thought, that worked well, what about the square root of i? Surely that must be yet a new kind of number. I just had to check that (a+bi)^2 can never be i. But whoa, you can solve that! (sqrt(2)/2) + (sqrt(2)/2)i does the trick. I was kind of bowled over by this. i, you sneaky bastard — you anticipated my next move and got ahead of me! I had no idea what “algebraically closed” meant, or anything like that. But it was one of my first experience of the incredible power of the right definition. Once the definition is right, you can just do everything.
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Quomodocumque 
My memory of figuring out mathematics as a teenager is being really bored at my grandparents’ place, realizing that one could use telescoping series to figure out the sum of the first n k-th powers, and doing some long hand calculations of the actual polynomials. At some point I got a Lotus spreadsheet to calculate what I now know are appropriate transforms of the first 20 or so Bernoulli numbers.
I wonder if someone can take these memories and predict that I turned out (roughly speaking) an algebraic combinatorialist and you turned out (roughly speaking) an arithmetic geometer.
Interestingly Deligne seems to have had precisely the same thought as documented in this interview: https://www.simonsfoundation.org/science_lives_video/pierre-deligne/
I suspect this may be a common thing, I remember doing this back in middle school as well.
I just hit the sqrt button on my TI-83 and got .70710678 + .70710678i. Figured that decimal had some meaning, so on a hunch I squared it and got .4999999. Only then did I formally verify that sqrt(1/2) + sqrt(1/2)i works.
@Kevin I believe the four color theorem was proved similarly, with a calculator proof assistant.
Hey! That’s *my* childhood memory!