It's just a mathematical tool1, which is especially useful to portray some kinds of vectors and sine waves. It doesn't have a real "meaning" here, and we can do fine whthout it (calculations just become more tedious)
For example, in circuit analysis, we convert the coordinate pair $(V,\phi)$ (which isn't linear; adding two voltages need not add their phases) into the coordinate pair $(V\cos\phi,V\sin\phi)$, which is linear. Now, instead of writing it as a coordinate pair, we write it as a complex number (which is essentially just a pair of numbers) and then notice that we can extend this to the concept of impedance and even be able to divide by these numbers and get the correct answer. Writing voltage as $(V\cos\phi)\hat i +(V\sin\phi)\hat j$ would still give a linear system, but we can't divide by vectors so it's limited in application.
In quantum mechanics, the use is similar -- we are expressing a linear quantity with a phase. Again, this is a nonlinear coordinate pair, but rewriting it as a complex number gives us a linear coordinate pair that plays nice with multiplication and division. When we bring linear algebra into it, one sees that the representation is even more apt.
Again, it's just a tool or a representation. It has no real meaning.
1. As @MBN mentioned below, real numbers are a mathematical tool, too. We can't "directly" measure real numbers, either.