An $N$-qubit system keeps $2^N$ pieces of information or should we say $2^{N+1}$? Say my quantum register is of size $3$ qubits. This means I'll need $2^3$ complex
numbers to describe all of its possible arrangements.  But each complex number requires $2$ real numbers, so maybe I could say I'll need $2^{3}\times2 = 2^4$ real numbers.  Is this correct?

Granted, a complex number is a number.  But when we implement a complex number on any classical computer, we always reserve two numbers for it.  A complex number is a number, but down to computer technology it really is a pair of numbers.  When it comes to storing information, the matter is more about technology than abstract science.

I know in complexity theory it doesn't matter if $2^{N}$ or $2^{N+1}$, but I haven't seen any mention of this little detail anywhere and I've been reading every introduction on quantum computing I can find.  Please do mention any source that you might know that has touched on this little technicality.  Thank you!
You are right to say that every complex number (that represents an quantum amplitude) is represented by a pair of real number on a classical computer. 
But, in my opinion, you are confusing a simulation of a real life quantum system
with a classical simulation.


In a classical simulation of a real life quantum system (say a quantum circuit that has not been built) we would associate a pair of real number-for representing a complex number that is a quantum amplitude of that quantum circuit -as you pointed out.
But if we actually build the quantum circuit in real life (rather than simulating it on a classical computer) then those complex amplitudes are actually an intrinsic part of the qubit (which may be implemented by ions, or photons)-because quantum states occur naturally-this is just basic physics. 


Another thing to know is that the "amplitude model" of reality is based on the postulates of quantum mechanics.
You are right to say that every complex number (that represents an quantum amplitude) is represented by a pair of real number on a classical computer. 
But, in my opinion, you are confusing a simulation of a real life quantum system
with a classical simulation.


In a classical simulation of a real life quantum system (say a quantum circuit that has not been built) we would associate a pair of real number-for representing a complex number that is a quantum amplitude of that quantum circuit -as you pointed out.
But if we actually build the quantum circuit in real life (rather than simulating it on a classical computer) then those complex amplitudes are actually an intrinsic part of the qubit (which may be implemented by ions, or photons)-because quantum states occur naturally-this is just basic physics. 


Another thing to know is that the "amplitude model" of reality is based on the postulates of quantum mechanics.