Why do complex numbers freak some people out?

 An interesting recent paper is discussed at Backreaction.  Go watch the video, it's only about 10 minutes.  I'll wait.

OK, got all that?  Now let's count: one potato, two potato, three potato, four.  I can show you any positive integer number of whole potatoes.  How about zero potatoes?  Hmm then, let's take them all off the desk, and now you've got zero potatoes.  At the same time you've also got zero apples, eggplants, or aardvarks on the table too.  Does this mean that counting down to zero magically transforms potatoes into any of the above?  No, of course not.  But it does mean that zero is an interesting abstraction all in and of itself.  How is this any more spooky than imaginary numbers?

Now let's take it one step farther: minus two potatoes, minus one potato, zero potato(es), one potato, two potatoes, etc.  Not too spooky, negative numbers are taught somewhere around second grade, so they seem kind of normal.  But have you ever seen a negative potato?  No, me neither.  Now I could owe you a potato, and a truckload of negative potatoes could be my balance down at the First National Bank of Produce and Vegetables.  In that sense a negative potato has some meaning, and it may be a useful abstraction and means of bookkeeping.  But you can't set an actual negative potato on my desk and say "Ah-ha!  Now you now owe me a potato!"

Similarly, there's a consistent algebra built up around complex and imaginary numbers (go watch the damn video if you haven't already), and it is very useful, say in any field involving wave mechanics among other uses.  Why is this any more spooky than negative numbers indicating what somebody owes to his friendly local loan shark?  (kneecaps aside)

Notice that this is somewhat close to the "shut up and calculate" attitude mentioned in the video, but it is not quite the same.  It's more of "understand your abstractions and the associated mathematics, then play them to the hilt" that I'm talking about here.  Either that, or I badly missed something back in second grade mathematics.  Back to the question "can you do wave mechanics without resorting to imaginary numbers?"  No, of course not.  It seems strange to me that this worries some people.  OTOH, it does take a little working through at first, but nobody ever said math is easy.

OK that was all kind of heavy.  Here, have a funny yet related cartoon.