Is there any reason to think that such crash would make the companies contained in the index go bankrupt?
Not by itself. A company doesn't care about the price of their own stock unless they're buying or selling it. If the company has enough money in their balance sheet they're not affected by their stock price.
If not, wouldn't the index eventually recover itself?
I don't understand how "recover" can ever be the right word. Bubble implies that the valuation is significantly higher than it should be. The bubble bursting would drop the valuation back to an acceptable range. Yes it would likely go back up to where it was eventually by the virtue of the stock market on average going up.
Would this hypothetical bubble affect equally both equity and bond indexes?
Bonds have a set value. The total payout of any bond you can buy is already set. The price of the bond is set by what buyers value that future money in today's money. That means bond prices tend to be fairly stable.