Using complex expressions in dealing with AC circuits is just a convenient way of representing impedance. The inductor and capacitor act differently in response to the alternating voltage. If they acted just like a linear resistance does (i.e., in phase with the voltage), there would be no need for complex numbers. Z is used to signify impedance, which is just a sort of resistance. So we have in a typical circuit R=resistance (which is in phase with the voltage), Xc=capacitive reactance (lags behind the voltage by 90° or a quarter cycle) and XL=inductive reactance (leads by 90°). Lagging and leading is a phase shift because AC voltage is sinusoidal.
z=R+(XL-Xc)j where j represents the imaginary part of the complex number z. All this means is that z is a type of vector with horizontal (real) component R and vertical (imaginary) component XL-Xc. The capacitive reactance acts in a negative vertical direction while inductive reactance acts in a positive vertical direction.
The impedance (dynamic resistance) is measured by working out |z|, the modulus of z, Z=√(R^2+(XL-Xc)^2). And we get the phase shift by working out arctan((XL-Xc)/R). The letter j is used in electrical engineering instead of i (imaginary square root of -1) because i can be confused with current, also represented by i. Also, z is used in complex algebra to represent a complex number and in electrical engineering Z usually means impedance. Z is the magnitude of the vector or complex number z.
The algebra of complex numbers is conveniently used to help us find the effective values of components in AC circuits.
I hope this helps.