A qubit is a unit of information, so the information included in one qubit is exactly one qubit. For most purposes, the information may be identified with the information of one bit. We call it a "quantum bit" because the two possibilities may be combined into arbitrary complex combinations such as $a|0\rangle + b|0\rangle$.
However, the complex amplitudes $a,b$ cannot be measured - at least not by a single measurement. You can measure whether the system described by the qubit is found in the state $|0\rangle$ or $|1\rangle$ (and you may also do similar measurements with any orthogonal basis of the 2-dimensional Hilbert space). If you do so, you obtain either 0 or 1, nothing else. You obtain 0 with the probability $|a|^2$ and 1 with the probability $|b|^2$.
Even if you choose another basis and make the measurement, you get just one bit of information from the measurement. More precisely, you usually get less than one bit - namely $-p_a\ln(p_a)-p_b\ln(p_b)$ where $p_{a}=|a|^2$ and similarly for $b$. The expression is maximized for $p_a=p_b=1/2$ where its value is $\ln(2)$ of information (in "nats") which is called one bit. For asymmetric choices of $p_a,p_b$, we get less than one bit of information from the measurement.
If you repeat the same experiment many times, or $K$ times, you may measure the amplitudes with the relative error of $1/\sqrt{K}$. However, it's the very point of quantum computation - the discipline where the notion of "qubit" actually becomes useful - that you only want to run the quantum algorithm once and get the result. If you needed to run the quantum algorithm many times, to measure the amplitudes, all the magic exponential speedup of quantum computers would be gone.
Moreover, even if you repeatedly try to measure the amplitudes, you won't find the equivalent of four real numbers. First of all, the wave function of the qubit has to be normalized so that the total probability of both/all possibilities is equal to 100 percent. It means that
$$|a|^2+|b|^2=1.$$
Moreover, the change of the overall phase i.e. the transformation
$$(a,b)\to (a\,\exp(i\phi),b\,\exp(i\phi))$$
is unphysical because the overall phase is unmeasurable by any tools, even in principle. So in fact, the number of measurable information - if you allow repeated measurements of the system in the same initial state - is not four real numbers but just two real numbers. Without a loss of generality, you may write
$$(a,b) = (A,e^{iB} \sqrt{1-A^2})$$
where $A,B$ are two real non-negative parameters. $A$ is between $0$ and $1$ while $B$ is between $0$ and $2\pi$.
So I said that it's fallacious to imagine that the amplitudes $a,b$ are classical numbers. They're not classical numbers in any sense - it's always the same debate about the wave function that most laymen always want to imagine as a "real classical wave". It's not a real classical wave. It's just a tool to predict probabilities. And one qubit is not the same thing as a "large number of classical bits". Instead, it is one classical bit that has certain qualitatively new properties. As long as you will try to imagine that quantum mechanics is the same thing as classical physics, just with some bigger objects, you will misunderstand the essence of quantum physics. Quantum physics is qualitatively different than anything we know from the (seemingly) classical world.
Alice, Bob, and photon
If Alice and Bob send signals to one another and the only information they can adjust is the timing of one photon, then the information carried by this timing is simply $\ln(N)/\ln(2)$ bits where $N$ is the number of "moments" or "intervals" that they can distinguish. If you constrain them by what they're allowed to measure, then it's meaningless to talk about quantum bits.
What they send to each other by the timing of the photon is some ordinary classical information. You would have to allow Alice and Bob to measure the interference of various possibilities - various timings - to turn them into a quantum computer, and then it could make some sense to talk about "qubits". However, as you designed it, the information is classical and the unit should be called one bit.