"j" notation is not strictly necessary, but you must do _something_ to account for the energy stored and released from the reactive components. using the concept of 'phase angle' and "j" notation is a method for doing this accounting that works quite well when you are working at a single frequency with fixed value components.
Another technique that you could use to figure out these sort of circuits is to calculate how the current changes with time, using equations that have time components (eg V*sine(t) for voltage, C*dV/dt for the current through the capacitor, etc) . The "V/I characteristic" of components ("resistance" generalized) changes with time, so V = I * V/I (Ohm's law) quickly becomes a pretty serious differential equation. This sort of approach is generally not necessary for 'human' calculations, but is quite often used in computer simulations. Also the basic concepts of how a capacitor works are more easily understood in terms of thinking of change in charge over time, so it is a nifty exercise to go from I=C*dV/dt (and the other associated equations) to phase angle and reactance notation.