Tesla bulls and bears can surely agree on one thing: Chief Executive Elon Musk is capable of making the surreal seem mundane.

Tuesday brought more of the electric-auto maker’s unique brand of routine chaos: A message from Mr. Musk’s Twitter account announced he is “considering” taking Tesla private at $420 a share and has secured funding to do so. Shares rocketed 11% and are now just below their record.

But...