Elon Musk went full-on corporate raider a week into his rolling clash with Twitter Inc., offering a $43 billion bid for the company and warning he might sell his stake in the service if rebuffed.

The Thursday offer was the latest in a will-he-or-won’t-he saga between the world’s richest person and the social-media service. The offer was at once serious—Mr. Musk disclosed it in a federal filing—and at the same time tinged with humor, as the offer was for $54.20 per share, a barely veiled reference.