I have $N$ objects, each having a performance function that depends on $|X_i|$, where $X_i$ is a "complex" random variable and $|\cdot|$ denotes the module of a complex number. The dependence is such that the higher $|X_i|$, the higher the performance of object $i$.
I want to create a PMF such that the probability of picking the $i^{th}$ object depends on its performance, i.e., the higher $|X_i|$, the higher the probability of picking the $i^{th}$ object.
Is it reasonable to consider the following as the probability of picking the $i^{th}$ object: $$ p_i = \frac{|X_i|}{\sum_{j=1}^N |X_j|} $$
I'm novice in probability so my concern is that while $p_i \in [0,1]$ and $\sum_{i=1}^N p_i = 1$, $p_i$ seems to be random and not a fixed number. Is it still a valid to consider this PMF? otherwise, what can I do to make it a proper PMF?
I assume you are looking for a random variable that, conditioning on $X_{1},\dots,X_{n}$, takes the value $|X_{i}|$ with probability $p_{i}=\frac{|X_{i}|}{\sum_{j}{|X_{j}|}}$. This sort of object indeed exists. We can find a realization this sort of random variable by taking a uniform random variable $U$ that is independent from $X_{1},\dots,X_{n}$, then take $$Y=\sum_{j}|X_{j}|\chi_{I_{j}}(U)$$ where $I_{i}$ is the interval $\left[\frac{\sum\limits_{j< i}|X_{j}|}{\sum\limits_{j}|X_{j}|},\frac{\sum\limits_{j\leq i}|X_{j}|}{\sum\limits_{j}|X_{j}|}\right]$ and $\chi_{I_{j}}$ is it's indicator function. Notice that for each fixed value of $X_{1},\dots,X_{n}$ we are picking $|X_{i}|$ with the desired probability.
Of course this is mostly a formality, as we can usually avoid working with the explicit realization by using conditional expectation.
Take $$P(Y\leq t)=E(E(\chi_{\leq t}(Y)|X))$$ Where we writing $X$ as the entire $(X_{1},\dots,X_{n})$ vector. Now conditionally on $X$, $Y$ takes the value $|X_{i}|$ with probability $p_{i}$, and $Y\leq t$ so long we pick an $i$ so that $|X_{i}|\leq t$. Put in other words, to know the expected value of $\chi_{\leq t}(Y)$ we have to count how many of the $|X_{i}|$ are no greater than $t$ and weight by the probability of picking those $|X_{i}|$, that is $$E(\chi_{\leq t}(Y)|X)=\sum\limits_{i}\chi_{\leq t}(|X_{i}|)p_{i}=\sum\limits_{i}\frac{\chi_{\leq t}(|X_{i}|)|X_{i}|}{\sum\limits_{j}|X_{j}|}$$
So $$P(Y\leq t)=\sum\limits_{i}E\left(\frac{\chi_{\leq t}(|X_{i}|)|X_{i}|}{\sum\limits_{j}|X_{j}|}\right)$$
If you know the joint density of $X$ you could calculate the PMF like would for any expected value, no guarantees you'll reach anything nice.
If you are not familiar with conditional expectation arguments you may reach a similarity conclusion by applying Fubini/Tonelli to the explicit realization of $Y$ we had above (by first integrating in respect to $U$). Though I suggest you look into it if you are interested in this sort of "nested" probabilities.