To 'go under' generally means to die. But in this context, it definitely doesn't mean that the bear died.
Hypothetically, let's assume this is what happened.
The bear, who is known for his fondness of honey, decided to give himself a treat, off a functional honeycomb. As we know, bees work very hard for honey and would risk their lives to protect their hives. The bear attacks and knocks down the honeycomb, but gets attacked by the furious bees. Now, owing to his natural instincts, he knows that diving into water is the only solution to a bee ambush. Since bears are natural swimmers, he decides to dive into the water and hide elsewhere, until the bees have left. He swam across lake and pops his head up, to see if the bees have left.
So from this, 'under' would only mean 'dived underwater', or simple mean 'went into hiding'. He had dived to escaped from the bees and the bees are waiting for him, at the other side of the lake, where the bear had apparently 'gone missing', from the perspective of the bees.