Algebraic geometry began over the field of reals. What Apollonius of Perga did would be certainly qualified today as algebraic geometry: he classified real plane curves of second order and studied their intersections, tangents, etc. in very great detail.
Certainly Apollonius had neither real numbers nor the notion of degree, but nevertheless we was able to obtain many non-trivial results.  

Another great contribution of antiquity is Diophantus' arithmetic.
Diophantus' book contains no theorems, only numerical examples. But he really understood somehow some basic facts which nowadays belong to algebraic geometry (rational curves and surfaces, not only in dimension 1 but in higher dimensions, rational points on elliptic curves do really appear in his work for the first time. He probably even understood the addition law on a plane non-singular cubic). 
(Ref. I. G. Bashmakova, Diophantus and Diophantine equations, MAA, 1997, MR1483067).


The general solution of $x^2+y^2=z^2$ in integers certainly belongs to algebraic geometry over rationals. Probably it was known even before Diophantus.

Until the late 19th nobody realized that the work of Apollonius and Diophantus belongs to the same area of mathematics. I think this was clearly understood for the first time in the work of Poincare
Sur les proprietes arithmetiques de courbes algebriques (1901).

But one can also say that true algebraic geometry really begins with Decartes who clearly established the 
relation between curves in
the plane and equations $f(x,y)=0$, thus establishing the
notion of algebraic curve and degree. And this was mostly real algebraic geometry. One result of Descartes was his "Rule of Signs" which has been much generalized in real algebraic geometry (fewnomials theory). An outstanding achievement of this real algebraic geometry was Newton's classification of plane real cubics.
Besout theorem was also proved before introduction of complex numbers to algebraic geometry.

Complex algebraic geometry begins only in 19th century, first by introduction of complex elements to projective geometry.

To Dedekind and Weber belongs the discovery that theory of algebraic functions is really similar to the theory of algebraic numbers, in the sense that both study finite field extensions, and much of this theory can be exposed in a unified way. They were able to prove many results of Riemann on algebraic curves by purely algebraic methods.

References:

R. Dedekind and H. Weber, Theory of algebraic functions of one variable, AMS 2010. (Several different English translations are available on Internet).

A. Weil, Number Theory. An approach through history. From Hammurapi to Legendre. Birkhauser, 1984.