Solution:

Let $a = 20$ and $b = 25$ be the initial numbers of blue and purple diamonds, respectively, and let $c = 44$ be the number of times Isabella performs the operation. Suppose that at some point, the bag contains $x$ blue diamonds and $y$ purple diamonds, for $x + y = z$ total diamonds. After one step, the bag will have $z + 1$ diamonds. The expected change in the number of blue diamonds in this step is $(- x / z) + 1 = y / z$, and likewise this quantity for purple diamonds is $x / z$. Thus, the expected change in the difference between the number of blue and purple diamonds is $(y - x) / z$. Since this difference was initially $x - y$, the expected value of this difference is multiplied by $(z - 1) / z$ at each step (regardless of $x - y$). Since $z$ starts at $a + b$ and ends at $a + b + c$, the expected difference after $c$ operations is
$$(a - b)\cdot \prod_{z = a + b}^{a + b + c - 1}\frac{z - 1}{z} = \frac{(a + b - 1)(a - b)}{(a + b + c - 1)},$$
and as the total number of diamonds is $a + b + c$, the expected number of blue diamonds at the end is
$$\frac{a + b + c}{2} + \frac{(a + b - 1)(a - b)}{2(a + b + c - 1)}.$$
Plugging in $a = 20$, $b = 25$, and $c = 44$ gives us the answer, $\boxed{\frac{173}{4}}$.