Does a light cone look the same from all reference frames?

If there were a light cone centered at some point $P$, and you were to look at that light cone from different reference frames, would it change its shape? I know that points inside and outside the light cone would remain inside/outside of the light cone in every frame, but does the light cone itself shift?

If it does, how would it shift?

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The light cone is a collection of events that are lightlike-separated from $P$. This collection of points is the same in all reference frames because in special relativity the interval is invariant.
If you swept out a light cone from $P$ by having a source at $P$ emit a spherical electromagnetic wave and noting where the wavefront was at future times, you would get the same result in all reference frames - that the edge of the wavefront was a sphere centered on $P$ with radius $ct$, with $t$ the time since the pulse was emitted. This is true even if source at $P$ is moving at the time the wavefront is emitted.