<s>[INST] Series of short questions intended to clarify $p$-norm of vectors with complex elements Note: 
My central question is Q.2.B. I'd really like an answer to that question.
Context:
From [1], we find that
``Let $ p \geq 1$ be a real number.  The $p$-norm  of vectors $\mathbf{x} = (x_1, \ldots, x_n)$  is
$$ \left\| \mathbf{x} \right\| _p := \bigg( \sum_{i=1}^n \left| x_i \right| ^p \bigg) ^{1/p}.$$
Questions:
[Q.1.A]
Is the $1$-norm  of vector $\mathbf{x} = ([1 + i], [3 + 2\,i], [6-20\,i])$ given as follows:
\begin{align}
 \left\| \mathbf{x} \right\| _1 
= 
&
\bigg( \sum_{i=1}^3 \left| x_i \right| ^1 \bigg) ^{1/1}
\\
=
&
\bigg(   \left|  1 + i  \right| + \left|  3 + 2\,i   \right|+ \left|    6-20\,i  \right|\bigg) 
\\
=
&
\bigg(   \left|  1 \right|+ \left|i  \right| + \left|  3 \right|+ \left| 2\,i   \right|+ \left|    6 \right| + \left|- 20\,i  \right|\bigg) 
\\
=
&
\bigg(     1 + 1    +   3  +   2    +     6   +   20 \bigg) 
\\
=
& 
\, 33? 
\end{align}
[Q.1.B]
Or, is the $1$-norm  of vector $\mathbf{x} = ([1 + i], [3 + 2\,i], [6-20\,i])$ given as follows:
\begin{align}
 \left\| \mathbf{x} \right\| _1 
= 
&
\bigg( \sum_{i=1}^3 \left| x_i \right| ^1 \bigg) ^{1/1}
\\
=
&
\bigg(   \left|  1 + i  \right| + \left|  3 + 2\,i   \right|+ \left|    6-20\,i  \right|\bigg) 
\\
=
&
\bigg(   \sqrt{1^2 + 1^2}  +  \sqrt{3^2 + 2^2} +  \sqrt{6^2 + 20^2} \bigg) ?
\end{align}
[Q.2.A]
Is there a named norm such that the named norm  of vector $\mathbf{x} = ([1 + i], [3 + 2\,i], [6-20\,i])$ is equal to 33 (see Q.1.A)? 
[Q.2.b]
If there is a such a named norm, what is the name of the norm?
[Q.3]
Is the $2$-norm  of vector $\mathbf{x} = ([1 + i], [3 + 2\,i], [6-20\,i])$ given as follows:
\begin{align}
 \left\| \mathbf{x} \right\| _2 
= 
&
\bigg( \sum_{i=1}^3 \left| x_i \right| ^2 \bigg) ^{1/2}
\\
=
&
\sqrt{   \left|  1 + i  \right|^2 + \left|  3 + 2\,i   \right|^2+ \left|    6-20\,i  \right|^2} 
\\
=
&
\sqrt{   \left( \sqrt{ 1^2 + 1^2 }\right)^2 + \left( \sqrt{ 3^2 + 2^2 }\right)^2 + \left( \sqrt{ 6^2 + (-20)^2 }\right)^2}
\\
=
&
\sqrt{      1^2 + 1^2    +   3^2 + 2^2   +   6^2 + (-20)^2  }
\\
=
& 
\sqrt{      1 + 1    +   9 + 4   +   36  + 400  }
\\
=
& 
\sqrt{           451  }?
\end{align}
Bibliography
[1]
https://en.wikipedia.org/wiki/Norm_(mathematics)#Generalizations
 [/INST] The wording on your question seems odd (not sure what a 'named norm' means) but I feel your real question translates to see if
$$\rho(z_1,\dots,z_n)=\sum_{i=1}^n |\textrm{Re}(z_i)|+|\textrm{Im}(z_i)|$$
is a norm on $\mathbb{C}^n$. To see that this isn't true just consider $v=(1+i,0,\dots,0)$ and $\lambda=1-i$. In order to have $\rho$ be a norm it shoud satisfy $\rho(\lambda v)=|\lambda|\rho(v)$, but we actually have $$\rho(\lambda v)=\rho(2,0,\dots,0)=2$$
and $$|\lambda|\rho(v)=2\sqrt 2 $$
This is basically the argument you used in your answer but with a concrete example to show it doesn't always hold.
</s>