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In a finite field with prime cardinality, every element can be expressed (by sum or products) by using only 1, the unit of the field. In a finite field with non-prime cardinality, it is not possible anymore since 1 will only generate the smaller subfield of prime cardinality. So, you need another element (called a "generator" of the field) to generate the whole field. This is the z2 you got:

sage: F = GF(4)
sage: F.gen()
z2

If you want to use z2 in your computations, you should attribute it the Python name z2:

sage: z2
NameError: name 'z2' is not defined
sage: F.inject_variables()
Defining z2
sage: z2
z2
sage: z2^2
z2 + 1
sage: z2^3
1