What would measuring the position of an electron in an electronic superposition look like? In view of the Copenhagen interpretation, the state:
$$\Psi = \frac{1}{\sqrt{2}}|0⟩ + \frac{1}{\sqrt{2}}|1⟩$$
is interpreted as "when the system is measured, you have a 50% chance of finding the system in state $|0⟩$, and a 50% chance of finding the system in state $|1⟩$".
Lets now look at a hypothetical hydrogen-like atom with a single electron. We will say the state is $|0⟩$ when the electron is in the $1 s$ orbital, and the state is $|1⟩$ if the electron is in a $2p_z$ orbital. Let's also assume that there is a higher-lying $3d_{z^2}$ state. The transition $3d_{z^2} \leftarrow 2p_{z}$ is allowed, $3d_{z^2} \leftarrow 1s$ is forbidden, and $3d_{z^2}$ decays via a fluorescent channel, emitting light which we can detect.
The readout of the state is performed by shining light tuned to the $3d_{z^2} \leftarrow 2p_{z}$ transition and measuring the fluorescence. Readout of the state $\Psi$ should result in fluorescence half of the time, the state $|1⟩$ should always fluoresce, and the state $|0⟩$ should never fluoresce. This agrees well with the Copenhagen interpretation - the system doesn't really know if it is in the $1s$ or $2p_{z}$ state until the measurement is performed.

Spectroscopies have advanced to the point in which we can actually map the distribution of electrons in specific orbitals. We are also developing lasers with shorter and shorter pulses. It is not unreasonable to expect that in some years we will be able to perform pump-probe velocity map imaging experiments and actually see the electron density as it evolves in a superposition.
Let's say we apply a pump pulse to our hydrogen atom for long enough to bring the state into $\Psi$. After a very short amount of time t, we measure the position of the electron using velocity map imaging (or similar technique). We repeat this many times in order to obtain an electron distribution at time t.
According to the Copenhagen interpretation, as it is often described, the moment the measurement is made the wave function collapses into either $1 s$ or $2p_{z}$. Half of the electrons would then come form $1s$, and half of the electrons would come from $2p_{z}$. The observed electron density would equal  $$\frac{1}{2}|1s|^{2} + \frac{1}{2}|2p_{z}|^{2}$$
But this is not what the Schrödinger equation predicts! According to the Schrödinger equation, a state in a superposition evolves, in the rotating frame, as:
$$\Psi(t) = \frac{1}{\sqrt{2}}|0⟩ + \frac{1}{\sqrt{2}}|1⟩e^{-i\omega t}$$
where $\omega = \frac{E_{2p_{z}} - E_{1s}}{\hbar}$
This equation predicts that the result of such pump-probe experiment would show the density evolve as:
$$|\Psi(t)|^{2} = \frac{1}{2}1s^{2} + \frac{1}{2}1p_{z}^{2} + (1s \times 2p_{z})\cos(\omega t)$$
             
             
   
If my understanding is correct, the results of such velocity map imaging experiments would indeed show the superposition to be a dynamically changing wavefunction. The gif above shows what the isosurface of the electron density of the state $\Psi(t)$ looks as a function of time in the ultra-fast timescale (a few hundreds of attoseconds) according to the Schrödinger equation. 
If my understanding is correct, the Copenhagen interpretation leads to different predictions than basic quantum mechanics - and we may be able to test these predictions in a few years. However, the fact that the Copenhagen interpretation is so accepted makes me believe that I must be misunderstanding something. 
Is my analysis correct? And, if so, does it mean that the Copenhagen interpretation of a superposition disagrees with basic QM? 
 A: 
According to the Copenhagen interpretation, as it is often described, the moment the measurement is made the wave function collapses into either $1 s$ or $2p_{z}$. Half of the electrons would then come form $1s$, and half of the electrons would come from $2p_{z}$. 

You're misinterpreting the Copenhagen interpretation. As mentioned in the comments the Copenhagen interpretation can mean several different, non-compatible things to different people; you seem to understand some form of objective-collapse theory by that term, which is a pretty reasonable way to take the term. However, you're missing a crucial part of the formalism:

the moment an energy measurement is made the wave function collapses into either $1 s$ or $2p_{z}$.

If you perform a projective measurement on some observable that's not compatible with energy (like, say, position), then the collapse will occur on the eigenbasis of the observable being measured. Your misunderstanding here isn't so much due to a misapplication of the Copenhagen Interpretation but rather of the basic rules of QM.
For the position measurement in your thought experiment, your core intuition is correct - you'll measure something akin to

$$|\Psi(t)|^{2} = \frac{1}{2}1s^{2} + \frac{1}{2}1p_{z}^{2} + (1s \times 2p_{z})\cos(\omega t),$$

which does indeed look like the animation you've provided. (For more information on that one, see my answer to this previous question.) Though, that said, it's important to note that if you're doing time-resolved imaging of an oscillating charge density (like e.g. the Goulielmakis paper referenced in the previous answer) then you need to be extremely careful with exactly what observable is being accessed by your experiment.
It's all in the game, though - it shouldn't be long before the exact measurement you've proposed, or something very closely equivalent, hits the literature. Heck, I'd be willing to put down a modest sum of money on a measurement hitting the books by, say, 2025 at the latest.
