感谢两位的回答，我按照巴拿赫空间的思路研究了一下最后这样画了：

\documentclass[boder=5pt]{standalone}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usepackage{tikz-3dplot}

\begin{document}

\begin{tikzpicture}[>=stealth]
\begin{axis}[scale=1.5,line width=0.5pt,axis lines=center,
        ticks=none,
        enlargelimits=false,
        xmin=-3, xmax=3, ymin=-3, ymax=3, zmin=-3, zmax=3,
        xlabel=$x$, ylabel=$y$, zlabel=$z$,
        every axis x label/.style={
          at={(axis cs:3,0,0)},
          anchor=west,
        },
        every axis y label/.style={
          at={(axis cs:0,3,0)},
          anchor=west,
        },
        every axis z label/.style={
          at={(axis cs:0,0,3)},
          anchor=south,
        }]
\addplot3 [black,domain=-1:1,samples=20,samples y=0]({x},-1,{x^2-1});
\addplot3 [black,domain=-1:1,samples=20,samples y=0]({x},0,{x^2});
\addplot3 [black,domain=-1:1,samples=20,samples y=0]({x},1,{x^2-1});
\addplot3 [black,domain=-1:1,samples=20,samples y=0](-1,{x},{1-x^2});
\addplot3 [black,domain=-1:1,samples=20,samples y=0](0,{x},{-x^2});
\addplot3 [black,domain=-1:1,samples=20,samples y=0](1,{x},{1-x^2});
\addplot3 [black,domain=-1:1,samples=20,samples y=0]({x},{-1/2},{x^2-1/4});
\addplot3 [black,domain=-1:1,samples=20,samples y=0]({x},{1/2},{x^2-1/4});
\addplot3 [black,domain=-1:1,samples=20,samples y=0]({-1/2},{x},{1/4-x^2});
\addplot3 [black,domain=-1:1,samples=20,samples y=0]({1/2},{x},{1/4-x^2});
\end{axis}
\end{tikzpicture}

\end{document}

效果还成。

寄存器的画法也很优秀，不过有个小问题是这样画出来的马鞍面和我画的其他的圆锥曲面之类风格差异有点大，放在一起会有点违和，但还是要感谢提供新知识！