Ability to add alarms set for a particular date and time, independent of due date. Many of us don't use due dates at all when a task doesn't have a concrete deadline, only start dates.
Adding my vote to this one. I do not use start dates and I prefer to use due dates without time but often would like to schedule a reminder for a particular time of day.
This is one of my biggest needs, especially in relation to recurring tasks. If I need to "Review Sam's Status Update" on the 15th of each month, I create a recurring task and put an alarm on it. But if this month Sam has issues that we are dealing with, I can't review the update, then set a reminder to look at it again in a week without changing the alarm date for the ongoing recurring task. The work around is to create an additional task for any recurring task for which you need a special alarm, but that just adds redundant tasks (and notes) to the system.