  In this article, we generalize the arithmetic degree and its related theory
to dynamical systems defined over an arbitrary field $\mathbf{k}$ of
characteristic $0$. We first consider a dynamical system $(X,f)$ over a
finitely generated field $K$ over $\mathbb{Q}$, we introduce the arithmetic
degrees $\alpha(f,\cdot)$ for $\overline{K}$-points by using Moriwaki heights.
We study the arithmetic dynamical degree of $(X,f)$ and establish the relative
degree formula. The relative degree formula gives a proof of the fundamental
inequality, that is, the upper arithmetic degree $\overline{\alpha}(f,x)$ is
less than or equal to the first dynamical degree $\lambda_1(f)$ in this
setting. By taking spread-outs, we extend the definition of arithmetic degrees
to dynamical systems over the field $\mathbf k$. We demonstrate that our
definition is independent of the choice of the spread-out. Moreover, in this
setting, we prove certain special cases of the Kawaguchi-Silverman conjecture.
A main novelty of this paper is that, we give a characterization of arithmetic
degrees of "transcendental points" in the case $\mathbf{k}=\mathbb{C}$, from
which we deduce that $\alpha(f,x)=\lambda_1(f)$ for very general $x\in
X(\mathbb{C})$ when $f$ is an endomorphism.
