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One of the method may be using some Blur (Fast Gausian) nodes in the compositor.

First check the Transparent checkbox in the Film panel of the Render header.

Add a cylinder, subdivide and shape it. Give it the Emission shader. Render image (F12). Go to compositor and add some Blur (Fast Gaussian) nodes with the different bluring values. Use Alpha Over nodes to mix them. Use Color node to color the beam as you like (in my case I used it to color the glow blue).

Use Render Layers (How to render a very complex scene using Render Layers?) to add a background or your scene with Godzilla in it to the final render.

EDIT: You can add two cylinders and place them in two different render layers. Each cylinder may have its own deforming shapekeys to affect the beam transformations. You may then parent both cylinders to the deforming bones. Here's the animated result. I know it isn't exactly the same as in the movie you provided but I guess it's a good start for experiments.

One of the method may be using some Blur (Fast Gausian) nodes in the compositor.

First check the Transparent checkbox in the Film panel of the Render header.

Add a cylinder, subdivide and shape it. Give it the Emission shader. Render image (F12). Go to compositor and add some Blur (Fast Gaussian) nodes with the different bluring values. Use Alpha Over nodes to mix them. Use Color node to color the beam as you like (in my case I used it to color the glow blue).

Use Render Layers (How to render a very complex scene using Render Layers?) to add a background or your scene with Godzilla in it to the final render.

EDIT: You can add two cylinders and place them in two different render layers. Each cylinder may have its own deforming shapekeys to affect the beam transformations. You may then parent both cylinders to the deforming bones. Here's the animated result. I know it isn't exactly the same as in the movie you provided but I guess it's a good start for experiments.

One of the method may be using some Blur (Fast Gausian) nodes in the compositor.

First check the Transparent checkbox in the Film panel of the Render header.

Add a cylinder, subdivide and shape it. Give it the Emission shader. Render image (F12). Go to compositor and add some Blur (Fast Gaussian) nodes with the different bluring values. Use Alpha Over nodes to mix them. Use Color node to color the beam as you like (in my case I used it to color the glow blue).

Use Render Layers (How to render a very complex scene using Render Layers?) to add a background or your scene with Godzilla in it to the final render.

EDIT: You can add two cylinders and place them in two different render layers. Each cylinder may have its own deforming shapekeys to affect the beam transformations. You may then parent both cylinders to the deforming bones. Here's the animated result. I know it isn't exactly the same as in the movie you provided but I guess it's a good start for experiments.

One of the method may be using some Blur (Fast Gausian) nodes in the compositor.

First check the Transparent checkbox in the Film panel of the Render header.

Add a cylinder, subdivide and shape it. Give it the Emission shader. Render image (F12). Go to compositor and add some Blur (Fast Gaussian) nodes with the different bluring values. Use Alpha Over nodes to mix them. Use Color node to color the beam as you like (in my case I used it to color the glow blue).

Use Render Layers (How to render a very complex scene using Render Layers?) to add a background or your scene with Godzilla in it to the final render.

EDIT: You can add two cylinders and place them in two different render layers. Each cylinder may have its own deforming shapekeys to affect the beam transformations. You may then parent both cylinders to the deforming bones. Here's the animated result. I know it isn't exactly the same as in the movie you provided but I guess it's a good start for experiments.

One of the method may be using some Blur (Fast Gausian) nodes in the compositor.

First check the Transparent checkbox in the Film panel of the Render header.

Add a cylinder, subdivide and shape it. Give it the Emission shader. Render image (F12). Go to compositor and add some Blur (Fast Gaussian) nodes with the different bluring values. Use Alpha Over nodes to mix them. Use Color node to color the beam as you like (in my case I used it to color the glow blue).

Use Render Layers (How to render a very complex scene using Render Layers?) to add a background or your scene with Godzilla in it to the final render.

EDIT: You can add two cylinders and place them in two different render layers. Each cylinder may have its own deforming shapekeys to affect the beam transformations. You may then parent both cylinders to the deforming bones. Here's the animated result. I know it isn't exactly the same as in the movie you provided but I guess it's a good direction for experiments.

One of the method may be using some Blur (Fast Gausian) nodes in the compositor.

First check the Transparent checkbox in the Film panel of the Render header.

Add a cylinder, subdivide and shape it. Give it the Emission shader. Render image (F12). Go to compositor and add some Blur (Fast Gaussian) nodes with the different bluring values. Use Alpha Over nodes to mix them. Use Color node to color the beam as you like (in my case I used it to color the glow blue).

Use Render Layers (How to render a very complex scene using Render Layers?) to add a background or your scene with Godzilla in it to the final render.

EDIT: You can add two cylinders and place them in two different render layers. Each cylinder may have its own deforming shapekeys to affect the beam transformations. You may then parent both cylinders to the deforming bones. Here's the animated result. I know it isn't exactly the same as in the movie you provided but I guess it's a good start for experiments.