How are $L^P$ spaces defined for complex valued functions? I know that for real valued functions $f\in L^p(\mathbb R^N)$, $p\in [0,\infty)$ if the following integral converges:
$$\int_{\mathbb R^N}|f(x)|^p dx$$
How is it for complex valued functions (of real variable)? Do I take modulus or absolute value of real and imaginary part? Does it matter?

For a complex number $z$, the expression $|z|$ means the complex modulus of $z$. So for a complex-valued (measurable) function $f:{\bf R}^n\to{\bf C}$, it makes sense to ask if $f$ is such that 
$$
\int_{{\bf R}^n}|f(x)|^p\ dx<\infty\tag{1}
$$
which gives the definition of $L^p$ space of complex-valued functions. Note that $|f(x)|$ here means the complex modulus of the complex number $f(x)$. 
See also the definition of $L^p$ spaces in  https://www.encyclopediaofmath.org/index.php/Lp_spaces

Note that for any real numbers $a$ and $b$, 
$$
\max\{|a|,|b|\}
\leq|a+bi|
\leq |a|+|b|
\leq 2\cdot\max\{|a|,|b|\}
$$
So an equivalent condition to (1) could be
$$
\int |\text{Re}\,f(x)|^p\ dx<\infty\quad\text{and  }
\int |\text{Im}\,f(x)|^p\ dx<\infty\tag{2}
$$
