How to use custom png image marker with plot?

I don't believe matplotlib can customize markers like that. See here for the level of customization, which falls way short of what you need.

As an alternative, I've coded up this kludge which uses matplotlib.image to place images at the line point locations.

import matplotlib.pyplot as plt
from matplotlib import image

# constant
dpi = 72
path = 'smile.png'
# read in our png file
im = image.imread(path)
image_size = im.shape[1], im.shape[0]

fig = plt.figure(dpi=dpi)
ax = fig.add_subplot(111)
# plot our line with transparent markers, and markersize the size of our image
line, = ax.plot((1,2,3,4),(1,2,3,4),"bo",mfc="None",mec="None",markersize=image_size[0] * (dpi/ 96))
# we need to make the frame transparent so the image can be seen
# only in trunk can you put the image on top of the plot, see this link:
# http://www.mail-archive.com/matplotlib-users@lists.sourceforge.net/msg14534.html
ax.patch.set_alpha(0)
ax.set_xlim((0,5))
ax.set_ylim((0,5))

# translate point positions to pixel positions
# figimage needs pixels not points
line._transform_path()
path, affine = line._transformed_path.get_transformed_points_and_affine()
path = affine.transform_path(path)
for pixelPoint in path.vertices:
    # place image at point, centering it
    fig.figimage(im,pixelPoint[0]-image_size[0]/2,pixelPoint[1]-image_size[1]/2,origin="upper")

plt.show()

Produces:

The other answer may lead to problems when resizing the figure. Here is a different approach, positionning the images inside annotation boxes, which are anchored in data coordinates.

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.offsetbox import OffsetImage, AnnotationBbox

path = "https://upload.wikimedia.org/wikipedia/commons/b/b5/Tango-example_icons.png"
image = plt.imread(path)[116:116+30, 236:236+30]

x = np.arange(10)
y = np.random.rand(10)

fig, ax = plt.subplots()
ax.plot(x,y)

def plot_images(x, y, image, ax=None):
    ax = ax or plt.gca()

    for xi, yi in zip(x,y):
        im = OffsetImage(image, zoom=72/ax.figure.dpi)
        im.image.axes = ax

        ab = AnnotationBbox(im, (xi,yi), frameon=False, pad=0.0,)

        ax.add_artist(ab)

plot_images(x, y, image, ax=ax)

plt.show()