Alternative to numpy's linalg.eig?
So if scikit's third eigenvector is (a,-b,-c,-d) then mine is (-a,b,c,d).

That's completely normal. If v is an eigenvector of a matrix, then -v is an eigenvector with the same eigenvalue.
I have written a simple PCA code that calculates the covariance matrix and then uses linalg.eig on that covariance matrix to find the principal components. When I use scikit's PCA for three principal components I get almost the equivalent result. My PCA function outputs the third column of transformed data with flipped signs to what scikit's PCA function does. Now I think there is a higher probability that scikit's built-in PCA is correct than to assume that my code is correct. I have noticed that the third principal component/eigenvector has flipped signs in my case. So if scikit's third eigenvector is (a,-b,-c,-d) then mine is (-a,b,c,d). I might a bit shabby in my linear algebra, but I assume those are different results. The way I arrive at my eigenvectors is by computing the eigenvectors and eigenvalues of the covariance matrix using linalg.eig. I would gladly try to find eigenvectors by hand, but doing that for a 4x4 matrix (I am using iris data set) is not fun.
Iris data set has 4 dimensions, so at most I can run PCA for 4 components. When I run for one component, the results are equivalent. When I run for 2, also equivalent. For three, as I said, my function outputs flipped signs in the third column. When I run for four, again signs are flipped in the third column and all other columns are fine. I am afraid I cannot provide the code for this. This is a project, kind of.