Do American Express cards have a similar authentication?
For Amex cards, the 'extra level of authentication' will work slightly differently. Amex uses the AVS-Address Verification System.
On the payment page on BookMyShow, you will be asked to enter your billing pin code. This will be checked against your card statement the bank has on its records.
If the pin code doesn't match your payment will be rejected. If you don't remember your exact billing pin code, you can confirm it from your Credit Card statement.