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I'm not quite certain what it is exactly that you're asking, but the wavefunction Ψ is complex-valued, so its amplitude is related to its product of it and its conjugate Ψ*. If I understand you correctly, you're wondering why the probability amplitude is the integral of ΨΨ* = |Ψ|² rather than sqrt[ΨΨ*] = |Ψ|. The reason for this that |Ψ|² has a corresponding probability current that satisfies Gauss' integral, whereas |Ψ| does not, i.e., |Ψ|² is conserved. This is not something peculiar to quantum-mechanical probability waves; ordinary optical interference involves waves of the electric field E and magnetic field B and yet the intensity varies as E² and B² instead. As a mathematical analogy, think of complex dot product x·y = x_k* y_k = |x||y|cos θ, and so that Ψ = aφ_1 + bφ_2, a≥0, b≥0, has ΨΨ* = a²|φ_1|² + ab[φ_1*φ_2 + φ_1φ_2*] + b²|φ_2|². The first and last terms are clearly the lone probability densities of first-slit and second-slit events, while the middle term 2ab Re[φ_1*φ_2] can be thought of as a measure of the phase difference between them, which determines constructive or destructive interference.

Without a bit more context, it's hard to interpret the situation correctly, but it seems to me that your book is talking about Feynman sum-over-histories approach to quantum mechanics. I'm not certain about the level your book is at, but since you've wondered about how this works in complex numbers, I'm going to work with that. I'm simplifying, since I really should be talking about probability densities and integrals, but this should suffice. A lot of this can be made much more precise.

The wavefunction φ is a complex wave whose absolute-square |φ|² = φφ* represents probability, where * is the complex-conjugate operator (u+iv→u-iv). Put this as an axiom for the moment; it has some substantial justification, but it is not simple. Your book most likely refers to |φ| as the amplitude of the wavefunction, so that the probability is its square (I've more often seen amplitude defined as a complex number in this case, but your book probably simplifies matters for pedagogical reasons). Just like ordinary waves, they combine according to the principle of superposition, i.e., Ψ = φ_1+φ_2. Using a bit of algebra, you can see that the probability density of the result is |Ψ|² = ΨΨ* = [φ_1+φ_2][φ_1*+φ_2*] = φ_1φ_1* + [φ_1φ_2*+φ_2φ_1*] + φ_2φ_2* = |φ_1|² + [...] + |φ_2|², i.e., the combined probability is the sum of the squares of the amplitudes plus some middle term. One of Feynman's rules is analogous to saying that the contribution of this term must be zero for distinguishable histories, i.e., they're in some sense orthogonal to each other, which is why I brought up the complex dot product as an analogy. And that's essentially what your book is saying--the combined probability is the sum of the squares of the amplitudes.

Sorry, I don't have a copy on me, but Feynman's rules are rather axiomatic in the sense that it takes a quite bit of work to show that they give equivalent results (on the other hand, they're in many cases simpler to work with). As per your original scenario, if there is no detector on either slit (no D1), then the particle going through either slit is indistinguishable from it going through the other, so the wavefunctions superpose--add then take absolute-square to get the probability of detection by a detector after both slits (x). If D1 is present, then there will be decoherence (under Copenhagen interpretation, wavefunction collapse), so that the system will be almost classical--probability of detection (by x) will be the direct sum of the probabilities of the particles going through each slit individually and winding up at x, i.e., sum of the absolute-squares. In Feynman's terms, the presence of D1 makes the alternative histories of the particles distinguishable.

Ah. Given your OP, I thought your question was more directed toward the mathematics and complex numbers side of the issue, rather than concpetual, so I directed my initial posts to that instead. In any case, I'm glad that everything worked out.