A Quarterstaff is not an implement.
A Quarterstaff is of the weapon group 'staff'. It is a staff by the definition of 'staff'.
Those that use staffs as implements find that the quarterstaff, which is a staff, is completely a staff.
Or are you saying a staff is not a staff?
A Staff is an implement, which can be used as a quarterstaff as a weapon.
Staff > Quarterstaff
not
Staff = Quarterstaff
or
Quarterstaff > Staff.
I don't think if you cannot read the weapon table and see CLEARLY that a Quarterstaff is a staff, then you cannot be an expert on this subject. It's... just not correct.
So yes, you can use a staff as a double weapon as, when used as a weapon, it functions as a quarterstaff. However, no 'weapony' effects transfer to it's implentitude.
A staff is a double weapon. Double weapons are wielded in each hand. You are therefore wielding a staff in each hand. Therefore it satisfies DIS...
...except that that staff's enhancement bonus has the same name as that staff's enhancement bonus. -That- is the reason they don't stack.
However, pre the 'same-name game element' rule, they stacked.
This isn't difficult to understand.
Moreover, DIS says directly, explicitly, verbatim, that you are adding the off-handed implement's enhancement bonus.
It's not saying 'A bonus equal to its enhancement bonus.' It says, explicitly that it is that enhancement bonus that you add. Are you saying that you can add the off-hand implement's enhancement bonus without actually adding the off-hand implement's enhancement bonus? Is that possible?
Please explain how.
You need two meet 2 criteria for DIS to work. 2 implements equipped and the implements need to be MAGICAL. A staff is a single implement, not two. So that is problem one. Second, using the optional rules for Inherent Bonus does not make a non-magical implement into a magical one. So you do not meet the criteria for DIS to work on both counts.
In the first count, you need to wield a magic implement in both hands. That is what DIS says. While, I agree that if you're wielding two implements, you ARE likely wielding an implement in both hands, you are not necessarily wielding an implement in both hands, nor is it required to wield two implements to wield an implement in both hands. The two are not logically equivalent.
Example:
If you are wielding a holy symbol and a staff, you do not satisfied Dual Implement Spellcaster.
If you are wielding a wand and a ki focus, you do not satisfy Dual Implement Spellcaster.
If you are wielding a rod, a holy symbol, and a ki focus, you do not satisfy Dual Implement Spellcaster.
On the second point, however, you are correct. The inherent bonus is an enhancement bonus to your powers; it is not the same as wielding a magic implement, or even two.