With MySQL, you have no choice: auto_increment fields *MUST* be UNIQUELY indexed. Meaning they are most commonly the PRIMARY KEY of the table. Not true of other DBs, but MySQL makes this mandatory. You can't even *create* the table if the auto_increment field isn't properly indexed.
But a bigger question is whether you should even *have* a field that is auto_increment.
If there is a *natural* primary key (e.g., phone number, social security number, anything guaranteed unique per record in the given table), you might be better off using it as the primary key and omitting the auto_increment field.
An optimist sees the glass as half full.
A pessimist sees the glass as half empty.
A realist drinks it no matter how much there is.