Storing connections would be a very bad idea. Databases usually have a connection limit, ie only 100 connections at one time.
If you stored open connections, leaving them open for each user... you could only have 100 sessions open before you need to either start deleteing sessions or changing MySQL settings to allow for more users.
Persistent connections are handled by PHP / Zend Engine for you and are the closest things to what you're asking for, use mysql_pconnect() instead of mysql_connect(). You DO NOT need to pass link identifiers for mysql_pconnect() to work, it will automatically check whether or not theres an open connection it can use.
In most tests I've done and seen done by others... mysql_pconnect() makes VERY little difference in script execution time, and in some cases just ties up connections for no reason.