Pattern-matching complex numbers as a + bi

I just started working with Mathematica and am toying with pattern matching. There may be something obvious I’m missing in this, but I can’t figure it out by myself.

I want to write down a function that takes a complex number as arguments. So f[1 + 2 I] should be a valid input, as well as f[a + b I]. I want, however, to make my function parse this as two numbers of the form a + bi, getting a and b by pattern matching. I made several attempts similar to this:

f[a_ + b_ I] := NSolve[a^2 + b^2 == 1/2 (1 + z), z]
SetAttributes[f, HoldAll]

(I guess the NSolve doesn’t matter in this case, but let it there in case it’s part of the issue.)

This doesn’t work as I planned. Any attempt to call it, like f[1 + 2 I], just echoes itself, but it does work fine when I call it with symbolic arguments, such as f[a + b I].

I guessed this should be due to some difference in the internal representation of symbolic expressions and complex numbers. Indeed, whenever I try to MatchQ[m + n I, a_ + b_ I], it says it’s True. But when I try the sorts of MatchQ[Unevaluated[2 + 3 I], a_ + b_ I], it’s False.

In trying to figure it out, I asked

FullForm[a + b I]
FullForm[Unevaluated[2 + 3 I]]
FullForm[a_ + b_ I]

and got

Plus[a,Times[Complex[0,1],b]]
Unevaluated[Plus[2,Times[3,[ImaginaryI]]]]
Plus[Pattern[a,Blank[]],Times[Complex[0,1],Pattern[b,Blank[]]]]

My questions are:

1. Shouldn’t the Plus[2,Times[3,[ImaginaryI]]] match with Plus[Pattern[a,Blank[]],Times[Complex[0,1],Pattern[b,Blank[]]]]?
2. What is the difference between [ImaginaryI] and Complex[0,1]? I know the first is a symbol as much as [Alpha] is, and I guess me asking for Unevaluated is preventing it from being cast as a Complex[0,1]. Probably this would be needed for the matching, but I don’t know a workaround.
3. Is there a better way to do what I’m attempting with my function?

Thanks!

f[a_. + b_ I | Complex[a_, b_]] := {a, b}