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I am looking for a book in projective geometry, using the apparatus of linear algebra, complex analysis, and, perhaps, modern algebra, in full. The counterexample is the Hartshorne's book on projective geometry that starts out with a list of axioms, as in high school geometry book. I'm looking for something in more Needham-like style.

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You could look at Chapters 4, 5, 6 of Berger's stupendous Geometry 1. There are plenty of calculations there, beautiful illustrations (including one by Escher), applications (to photography, optics) and links to algebraic topology: Hopf fibration, Boy's surface, simple connctedness, Veronese surface,...

He also carefully analyzes the canonical projectivization of an affine space and relates affine to projective groups. There is even a short discussion of projective space over the quaternions and over finite fields.

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It may be a bit dated but I think Linear Algebra and Projective Geometry by Baer is what you are looking for. I have a copy and it is very much geared towards the approach you were hoping to find.

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