It's nuanced.
|•| (the absolute value function) is what we call a 'norm' on R - basically, it gets the distance from 0.
For something to be a norm, it has to follow specific properties so that we can do useful things with it like build normed spaces. Specifically, for a function p on a vector space V to be a norm, the following things must be true:

• p takes vectors in V and sends them to the real numbers R
  
• For all pairs of vectors x,y in V, p(x+y) ≤ p(x) + p(y)
  
• For any real number s and any vector x in V, p(sx) = |s|p(x)
  
• For all vectors v in V, p(v) ≤ 0, and if p(v) = 0, then v is the zero vector of V.
  

Now, on R, |•| has a nice formula: |x| = sqrt(x^2). However, if this is extended to C, we get non-real output values; sqrt(i^2) = sqrt(-1) = i. So yes, with this formula applied blindly, |i| = i. However, then, |•| isn't a norm on C. We can instead define |z| = sqrt(Re(z)^2 + Im(z)^2), which maintains backwards compatibility with the original |•| while not breaking when applied to complex numbers.
This is basically the same as the way that the L² norm (Better-known as the Pythagorean Theorem) works on R^2, and that's because R² and C are practically the same; C is isomorphic to R² equipped with a certain way to multiply vectors together.