Remote vehicle ignition - code help.

If you ever get around to answering the question of how what that code does differs from what you want it to do, and post more code, please don't post code like that again.

    if (digitalRead(alternatorPin)==LOW) 
       {digitalWrite(ignitionPin,HIGH);}
       else
       {digitalWrite(ignitionPin,LOW);}

is way too hard to read or modify.

    if (digitalRead(alternatorPin)==LOW) 
    {
       digitalWrite(ignitionPin,HIGH);
    }
    else
    {
       digitalWrite(ignitionPin,LOW);
    }

is the same code with judicious white space, which is much easier to read (and see the logic of, at a glance).

There is nothing in that initial code that knows anything about "so the user gets to the car before the 15 minutes is up". If that is the problem, why not say so?