The formula:

(v|w) := v_1w_1 + 2v_2w_2

does NOT define an inner product space on the complex plane. Because the Hermitian Symmetry axiom fails:

(v|w) ≠ (w|v)*

Proof. (NOTE: (*) means conjugate)

(v|w) = v_1w_1 + 2v_2w_2

(w|v) = w*_1v_1 + 2w*_2v_2

(w|v)* = (w*_1v_1 + 2w*_2v_2)*
= w_1v*_1 + 2w_2v*_2
=v*_1w_1 + 2v*_2w_2
≠ (w|v)

Could someone just check this please.

Another thing...
If
(v|w) = v_1w*_1 + v_2w*_2
does
(w|v) = w_1v*_1 + w_2v*_2

if you're working in a two-dimensional Euclidean vector space then most everything you have stated in correct. The first line of your proof is incorrect, as (v,w) = w_1v_1* + w_2v_2*. Really the definition of an inner product over the complex field is that (v,w) = (w,v)*, and any defined inner product must have that property.

The first line of your proof is incorrect

How can this be incorrect? I was re-stating the question.

(v|w) = v_1w_1 + 2v_2w_2

I have to show that this is or is not an inner product space over the complex plane C².

Obviously it is not because the axiom fails. I was just wondering if anyone thought otherwise.

And I'll assume that if
(v|w) = v_1w*_1 + v_2w*_2
then
(w|v) = w_1v*_1 + w_2v*_2

When you interchange v and w in the inner product, you swap v and w in the formula and conjugate it right?

(w|v)* = (w_1v*_1 + w_2v*_2)* = v_1w*_1 + v_2w*_2 = (v|w)

Ok, I think I just answered it myself!