How to design a left shift register
One approach is to design the circuit as a regular two bit binary counter using a pair of flip-flops. Then you put some circuitry on the outputs that decodes the four counter states into the output sequence that you desire. Such decode functions actually already exist as standalone chips. 

Here is the state table related to this:

Binary 00 -> Decode 0001

Binary 01 -> Decode 0010

Binary 10 -> Decode 0100

Binary 11 -> Decode 1000

Binary 00 -> Decode 0001

.....

[and so on]

There are other ways to make your recirculating shift register but they often have the problem of getting unwanted states in to the output stream such that more than one bit would be set at a time. The binary counter followed by decoder does not have that problem.

You may find it interesting to know that there is a commonly available chip that performs the counter and decoder operation all in one IC chip. The CD74HC4017 is one that does this with 10 output pins.